Hitting forbidden induced subgraphs on bounded treewidth graphs

For a fixed graph $H$, the $H$-IS-Deletion problem asks, given a graph $G$, for the minimum size of a set $S \subseteq V(G)$ such that $G\setminus S$ does not contain $H$ as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph $H$, the smallest function $f_H(t)$ such that $H$-IS-Deletion can be solved in time $f_H(t) \cdot n^{O(1)}$ assuming the Exponential Time Hypothesis (ETH), where $t$ and $n$ denote the treewidth and the number of vertices of the input graph, respectively. We show that $f_H(t) = 2^{O(t^{h-2})}$ for every graph $H$ on $h \geq 3$ vertices, and that $f_H(t) = 2^{O(t)}$ if $H$ is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when $H$ deviates slightly from a clique, the function $f_H(t)$ suffers a sharp jump: if $H$ is obtained from a clique of size $h$ by removing one edge, then $f_H(t) = 2^{\Theta(t^{h-2})}$. We also show that $f_H(t) = 2^{\Omega(t^{h})}$ when $H=K_{h,h}$, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function $f_{C_4}(t)$ for the subgraph version. Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of $G$ is colored with some color from $V(H)$ and we require to hit only induced copies of $H$ with matching colors. In this case, we determine, under the ETH, the function $f_H(t)$ for every connected graph $H$ on $h$ vertices: if $h\leq 2$ the problem can be solved in polynomial time; if $h\geq 3$, $f_H(t) = 2^{\Theta(t)}$ if $H$ is a clique, and $f_H(t) = 2^{\Theta(t^{h-2})}$ otherwise.


Introduction
Motivated by the work of Cygan et al. [13], we also consider the colorful variant of the problem, where the input graph G comes equipped with a coloring σ : V (G) → V (H) and we are only interested in hitting induced subgraphs of G isomorphic to H such that their colors match. In this case, we first observe that essentially the same dynamic programming algorithm of the non-colored version (Theorem 3) yields the upper bound f H (t) = 2 O(t h−2 ) for every graph H on h ≥ 3 vertices. Again, our main contribution concerns lower bounds: we show (Theorem 16), by modifying appropriately the frame introduced for the non-colored version, that f H (t) = 2 Ω(t h−2 ) for every graph H having a connected component on h vertices that is not a clique. Since the case where H is a clique can also be easily solved in single-exponential time (Theorem 5), which can be shown (Theorem 18) to be optimal, it follows that if H is a connected graph on h ≥ 3 vertices, f H (t) = 2 Θ(t) if H is a clique, and f H (t) = 2 Θ(t h−2 ) otherwise. It is easy to see that the cases where |V (H)| ≤ 2 can be solved in polynomial time by computing a minimum vertex cover in a bipartite graph.
Organization. In Section 2 we provide some basic preliminaries and formally define the problems. In Section 3 we present the algorithms for both problems, and in Section 4 (resp. Section 5) we provide the lower bounds for the non-colored (resp. colored) version. Finally, we conclude the article in Section 6 with some open questions.

Preliminaries
Graphs and functions. We use standard graph-theoretic notation, and we refer the reader to [15] for any undefined notation. We will only consider undirected graphs without loops nor multiple edges, and we denote an edge between two vertices u and v by {u, v}. A subgraph H of a graph G is induced if H can be obtained from G by deleting vertices. A graph G is H-free if it does not contain any induced subgraph isomorphic to H. For a graph G and a set S ⊆ V (G), we use the notation G \ S := G[V (G) \ S]. A vertex v is complete (resp. anticomplete) to a set S ⊆ V (G) if v is adjacent (resp. not adjacent) to every vertex in S. A vertex set S of a connected graph G is a separator if G \ S is disconnected. We denote by ∆(G) (resp. ω(G)) the maximum vertex degree (resp. clique size) of a graph G. For an integer h ≥ 1, we denote by P h (resp. I h , K h ) the path (resp. independent set, clique) on h vertices, and by K h − e the graph obtained from K h by deleting one edge. For two integers a, b ≥ 1, we denote by K a,b the bipartite graph with parts of sizes a and b. We denote the disjoint union of two graphs G 1 and G 2 by G 1 + G 2 .
A graph property is hereditary if whenever it holds for a graph G, it holds for all its induced subgraphs as well. The open (resp. closed) neighborhood of a vertex v is denoted by N (v) (resp. N [v]). A vertex is simplicial if its (open or closed) neighborhood induces a clique. A graph G is chordal if it does not contain induced cycles of length at least four or, equivalently, if V (G) can be ordered v 1 , . . . , v n such that, for every 2 ≤ i ≤ n, vertex v i is simplicial in the subgraph of G induced by {v 1 , . . . , v i−1 }. Note that being chordal is a hereditary property.
Given a function f : A → B between two sets A and B and a subset A ⊆ A, we denote by f | A the restriction of f to A and by im(f ) the image of f , that is, im(f ) = {b ∈ B | ∃a ∈ A : f (a) = b}. For an integer k ≥ 1, we let [k] be the set containing all integers i with 1 ≤ i ≤ k.
Definition of the problems. Before formally defining the problems considered in this article, we introduce some terminology, mostly taken from [13]. Given a graph H, an H-coloring of a graph G is a function σ : V (G) → V (H). A homomorphism (resp. induced homomorphism from a graph H to a graph G is a function π : V (H) → V (G) such that {u, v} ∈ E(H) implies (resp. if and only if) {π(u), π(v)} ∈ E(G). When G is H-colored by a function σ, an (induced) σ-homomorphism from H to G is an (induced) homomorphism π from H to G with the additional property that every vertex is mapped to the appropriate color, that is, σ(π(a)) = a for every vertex a ∈ V (H). An (induced) H-subgraph of G is an (induced) injective homomorphism from H to G and, if G is H-colored by a function σ, an (induced) σ-H-subgraph of G is an (induced) injective σ-homomorphism from H to G. We say that a vertex set X ⊆ V (G) hits an (induced) σ-H-subgraph π if X ∩ π(V (H)) = ∅.
For a fixed graph H, the problems we consider in this article are defined as follows.

H-IS-Deletion
Input: A graph G. Output: The minimum size of a set X ⊆ V (G) that hits all induced H-subgraphs of G.
Colorful H-IS-Deletion Input: A graph G and an H-coloring σ of G. Output: The minimum size of a set X ⊆ V (G) that hits all induced σ-H-subgraphs of G.
The H-S-Deletion and Colorful H-S-Deletion problems are defined similarly, just by removing the word 'induced' from the above definitions. In the decision version of these problems, we are given a target budget k, and the objective is to decide whether there exists a hitting set of size at most k. Unless stated otherwise, we let n denote the number of vertices of input graph of the problem under consideration. When expressing the running time of an algorithm, we will sometimes use the O * (·) notation, which suppresses polynomial factors in the input size.

Tree decompositions.
A tree decomposition of a graph G is a pair D = (T, X ), where T is a tree and X = {X w | w ∈ V (T )} is a collection of subsets of V (G), called bags, such that: w∈V (T ) X w = V (G), for every edge {u, v} ∈ E, there is a w ∈ V (T ) such that {u, v} ⊆ X w , and for each {x, y, z} ⊆ V (T ) such that z lies on the unique path between x and y in T , X x ∩ X y ⊆ X z . We call the vertices of T nodes of D and the sets in X bags of D. The width of a tree decomposition D = (T, X ) is max w∈V (T ) |X w | − 1. The treewidth of a graph G, denoted by tw(G), is the smallest integer t such that there exists a tree decomposition of G of width at most t. We need to introduce nice tree decompositions, which will make the presentation of the algorithms much simpler.
Nice tree decompositions. Let D = (T, X ) be a tree decomposition of G, r be a vertex of T , and G = {G w | w ∈ V (T )} be a collection of subgraphs of G, indexed by the vertices of T . A triple (D, r, G) is a nice tree decomposition of G if the following conditions hold: X r = ∅ and G r = G, each node of D has at most two children in T , for each leaf ∈ V (T ), X = ∅ and G = (∅, ∅). Such an is called a leaf node, if w ∈ V (T ) has exactly one child w , then either The node w is called an introduce node and the vertex v in is the introduced vertex of X w , X w = X w \ {v out } for some v out ∈ X w and G w = G w . The node w is called a forget node node and v out is the forget vertex of X w . if w ∈ V (T ) has exactly two children w 1 and w 2 , then For each w ∈ V (T ), we denote by V w the set V (G w ). Given a tree decomposition, it is possible to transform it in polynomial time to a nice one of the same width [24]. Moreover, by Bodlaender et al. [7] we can find in time 2 O(tw) · n a tree decomposition of width O(tw) of any graph G with treewidth tw. Since the running time of our algorithms dominates this function, we may assume that a nice tree decomposition of width t = O(tw) is given along with the input.
Exponential Time Hypothesis. The Exponential Time Hypothesis (ETH) of Impagliazzo and Paturi [21] implies that the 3-Sat problem on n variables cannot be solved in time 2 o(n) . The Sparsification Lemma of Impagliazzo et al. [22] implies that if the ETH holds, then there is no algorithm solving a 3-Sat formula with n variables and m clauses in time 2 o(n+m) .
Using the terminology from Cygan et al. [13], a 3-Sat formula ϕ in conjunctive normal form is said to be clean if each variable of ϕ appears exactly three times, at least once positively and at least once negatively, and each clause of ϕ contains two or three literals and does not contain twice the same variable. Cygan et al. [13] observed the following useful lemma. Lemma 1 (Cygan et al. [13]). The existence of an algorithm in time 2 o(n) deciding whether a clean 3-Sat formula with n variables is satisfiable would violate the ETH.

Algorithms
In this section we present algorithms for H-IS-Deletion and Colorful H-IS-Deletion. We start in Subsection 3.1 with a general dynamic programming algorithm that solves H-IS-Deletion and Colorful H-IS-Deletion in time O * (2 O(t h−2 ) ) for any graph H on at least h ≥ 3 vertices. In Subsection 3.2 we focus on hitting cliques and independent sets.

A general dynamic programming algorithm
We present the algorithm for H-IS-Deletion, and then we discuss that essentially the same algorithm applies to Colorful H-IS-Deletion as well. Our algorithm to solve ) uses standard dynamic programming over a nice tree decomposition of the input graph; we refer the reader to [12] for a nice exposition. However, in order to achieve the claimed running time, we need to use a slightly non-trivial encoding in the tables, which we proceed to explain. Let |V (H)| = h and assume that we are given a nice tree decomposition of the input graph G such that its bags contain at most t vertices (in a tree decomposition of width t, the bags have size at most t + 1, but to simplify the exposition we assume that they have size at most t, which does not change the asymptotic complexity of the algorithm).
Intuitively, our algorithm proceeds as follows. At each bag X w of the nice tree decomposition of G, a state is indexed by the intersection of the desired hitting set constructed so far with the bag, and the collection of proper subgraphs of H that occur as induced subgraphs in the graph obtained from G w after removing the current solution. In order to be able to proceed with the dynamic programming routine while keeping the complement of the hitting set H-free, we need to remember how these proper subgraphs of H intersect with X w , and this is the most expensive part of the algorithm in terms of running time. We encode this collection of rooted subgraphs (where the "roots" correspond to the vertices in X w ) of H with an object H w that we call a rooted H-folio, inspired by similar encodings in the context of graph minors [1,3]. Since we need to remember proper subgraphs of H on at most h − 1 vertices, and we have up to t choices to root each of their vertices in the bag X w , the number of rooted proper subgraphs of H is at most t h−1 . Therefore, the number of rooted H-folios, each corresponding to a collection of rooted proper subgraphs of H, is bounded from above by 2 t h−1 . This encoding naturally leads to a dynamic programming algorithm to solve H-IS-Deletion in time O * (2 O(t h−1 ) ), where the hidden constants (but not the degree of the polynomial in n) may depend on H.
In order to further reduce the exponent to h − 2, we use the following trick inspired by the dynamic programming algorithm of Bonomo-Braberman et al. [9] to solve K 1,3 -IS-Deletion in time O * (2 O(t 2 ) ). The crucial observation is the following: the existence of proper induced subgraphs of H that are fully contained in the current bag X w can be checked locally within that bag, without needing to root their vertices. That is, we distinguish these local occurrences of proper induced subgraphs of H, and we encode them separately in H w , without rooting their vertices in X w . Note that the number of choices for those local occurrences depends only on H. In particular, since the proper subgraphs of H have at most h − 1 vertices, the previous observation implies that we never need to root exactly h − 1 vertices of an induced subgraph of H, since such occurrences would be fully contained in X w . This permits to improve the running time to O * (2 O(t h−2 ) ). The details follow.
Note that we may assume that H has at least three vertices, as otherwise it is a clique or an independent set, and then H-IS-Deletion can be solved in single-exponential time by the algorithms in Subsection 3.2. Proof. As discussed in Section 2, we may assume that we are given are nice tree decomposition (D, r, G) of G of width O(tw(G)). To simplify the exposition, suppose that the bags of D contain at most t vertices, and recall that this assumption does not change the asymptotic complexity of the algorithm.
At each bag X w of the given nice tree decomposition of G, a valid state of our dynamic programming table is indexed by the following two objects: A subsetŜ w ⊆ X w that corresponds to the intersection of the desired H-hitting set with the current bag. With this in mind, we say that a set A rooted H-folio H w , containing the following two collections of (rooted) proper induced subgraphs of H: The set L w of local occurrences of proper induced subgraphs of H, consisting of the set of proper induced subgraphs of H that occur in G[X w \Ŝ w ]. The set R w of rooted occurrences of proper induced subgraphs of H, consisting of a set of triples (H, R, ρ) whereH is a proper induced subgraph of H, R ⊆ V (H) is a set with 0 ≤ |R| ≤ h − 2 corresponding to the roots ofH in X w \Ŝ w , and ρ : R → X w \Ŝ w is an injective function that maps each vertex in R to its corresponding vertex in X w \Ŝ w .
Note that the number of local occurrences of proper induced subgraphs of H depends only on H, and that the number of tuples (H, R, ρ) of rooted occurrences is at most 2 h · 2 h−1 · t h−2 , and therefore the number of rooted H-folios is at most 2 O(t h−2 ) , as desired.
We say that the rooted H-folio of a subgraph G w ⊆ G w is H w if the local and rooted occurrences of induced subgraphs of H in G w correspond exactly to the collections L w and R w of H w , respectively. Our algorithm stores, for each state (Ŝ w , H w ) of a node w of a nice tree decomposition of G, the minimum size of a set S w ⊆ V w feasible forŜ w such that the rooted H-folio of G w \ S w is H w , or +∞ if such a set does not exist. We denote this value by opt(Ŝ w , H w ).
When r is the root of the nice tree decomposition, note that the solution of the H-IS-Deletion problem in G equals min Hr {opt(∅, H r )}, where H r = (L r , R r ) runs over all rooted H-folios such that L r = ∅ and R r contains the triples (H, ∅, ∅) for all proper induced subgraphsH of H.
We now show how these valid states and associated values can be computed recursively in a typical bottom-up fashion starting from the leaves, by distinguishing the distinct types of nodes in a nice tree decomposition. We let V (H) = {z 1 , . . . , z h }.
Introduce node. Let w be an introduce node with child w such that X w \ X w = {v}. For each valid state (Ŝ w , H w ) for w , with H w = (L w , R w ), we generate the following valid states for w, depending on whether v is included in the current partial hitting set in X w or not: (Ŝ w ∪ {v}, H w ). In this case, we just include v into the partial hitting set, hence the rooted H-folio remains the same. Therefore, opt(Ŝ w ∪ {v}, H w ) = opt(Ŝ w , H w ) + 1.
That is, since in this case v does not belong to the partial hitting set, we also add to R w any rooted occurrence that can be obtained from a rooted occurrence in R w by adding vertex v to the set of roots R to form a largerH. In this case we set opt(Ŝ w , H w ) = opt(Ŝ w , H w ).
Forget node. Let w be a forget node with child w such that X w \ X w = {v}. For each valid state (Ŝ w , H w ) for w , with H w = (L w , R w ), we generate the following valid states for w, depending on whether v ∈Ŝ w or not: If v ∈Ŝ w we add the state (Ŝ w \{v}, H w ) and we set opt(Ŝ w \{v}, H w ) = opt(Ŝ w , H w ). In this case, we just forget vertex v, which was in the solution, and H w remains the same. Otherwise, if v / ∈Ŝ w , we add the state (Ŝ w , H w ) where H w = (L w , R w ) is defined as follows: L w contains all the proper induced subgraphs of H that occur in G[X w \Ŝ w ]. Again, this set can be computed locally in time O(t h ). R w contains every triple (H, R, ρ) that can be constructed by any of the following two operations: * If there is a local occurrenceH ∈ L w such that v belongs to an inducedHsubgraph F of G[X w \Ŝ w ], we add to R w the triple (H, R, ρ) defined asH =H , where z is the vertex ofH mapped to v, and ρ mapping every vertex of R to their image in F . That is, if v was part of a local occurrence for node w , now this occurrence becomes a rooted one for node w, defined in the natural way. * Let (H , R , ρ ) ∈ R w be a rooted occurrence in R w . We distinguish two cases: , we add to to R w the triple (H, R, ρ) defined asH =H , R = R \ {ρ −1 (v)}, and ρ = ρ | R . That is, we just remove the forgotten vertex v from the root set of the corresponding rooted occurrence.
In this case we set opt(Ŝ w , H w ) = opt(Ŝ w , H w ).
Join node. Let w be a join node with children w 1 and w 2 . For each pair of valid states and R w is defined as follows. For every pair of rooted triples . That is, we just merge the rooted triples that coincide in X w = X w1 = X w2 , by taking the union of the corresponding subgraphs of H. Finally, for (Ŝ w1 , H w ) to be indeed a valid state for w, we have to check that an occurrence of H has not been created in those triples: if for some such a triple (H 1 ∪H 2 , R 1 , ρ 1 ) ∈ R w , we have thatH 1 ∪H 2 = H, we discard the state (Ŝ w1 , H w ) for w, and we move on to the next pair of valid states (Ŝ w1 , H w1 ) and (Ŝ w2 , H w2 ) for w 1 and w 2 , respectively.
If the state (Ŝ w1 , H w ) has not been discarded, we set In all cases, if distinct valid states of the child(ren) node(s) generate the same valid state (Ŝ w , H w ) at the current node w, we update opt(Ŝ w , H w ) to be the minimum among all the obtained values, as usual. This concludes the description of the algorithm, whose correctness follows from the definition of the tables and the fact that the solution of the H-IS-Deletion problem on G is computed at the root of the nice tree decomposition. Clearly, all the above operations can be performed at each node in time 2 O(t h−2 ) , and the proof is complete by taking into account that we may assume that the given nice tree decomposition has O(t · n) nodes [24].
A dynamic programming algorithm similar to the one provided in Theorem 2 can also solve the Colorful H-IS-Deletion problem in time 2 O(t h−2 ) · n for every graph H on h ≥ 3 vertices. Indeed, the algorithm remains basically the same, except that we have to keep track only of colorful copies of proper subgraphs of H, and to discard only the states in which a colorful occurrence of H appears. In order to do that, in the tables of the dynamic programming algorithm we just need to replace rooted H-folios by rooted σ-H-folios, defined in the natural way. Since the number of further computations at each node in order to verify that the colors match in the obtained rooted subgraphs of H is a function dominated by 2 O(t h−2 ) , we obtain the same asymptotic running time. We omit the details. It is easy to check that small adaptations of the algorithms of Theorems 2 and 3 also work for the (not necessarily induced) subgraph version of both problems. Nevertheless, the obtained running times never outperform those obtained by Cygan et al. [13] for those problems.

Hitting cliques and independent sets
The following folklore lemma follows easily from the definition of tree decomposition.

Lemma 4. Let G be a graph and let D be a tree decomposition of G. Then every clique of G is contained in some bag of D.
Note that if H is a clique, then the (Colorful) H-IS-Deletion problem is the same as the (Colorful) H-S-Deletion problem. Cygan et al. [13] observed that, by Lemma 4, in order to solve (Colorful) K h -IS-Deletion it is enough to do the following: store, for every bag of a (nice) tree decomposition of the input graph, the subset of vertices of the bag that belongs to the partial hitting set, and check locally within the bag that the remaining vertices do not induce a K h . A typical dynamic programming routine yields the following result 2 . The case where H is an independent set, which is NP-hard by [25], turns out to be more interesting. We proceed to present a single-exponential algorithm for I h -IS-Deletion, and we remark that this algorithm does not apply to the colorful version.
Note that I 2 -IS-Deletion is the dual problem of Maximum Clique, since a minimum I 2 -hitting set is the complement of a maximum clique. This duality together with Lemma 4 yield the following key insight: in any graph G, after the removal of an optimal solution of I 2 -IS-Deletion, all the remaining vertices are contained in a single bag of any tree decomposition of G. Our algorithm is based on a generalization of this property to any h ≥ 1, stated in Lemma 7, which gives an alternative way to exploit tree decompositions in order to solve the H-IS-Deletion problem.
We first need a technical lemma. A clique cover of a graph G is a collection of cliques of G that cover V (G), and its size is the number of cliques in the cover.

Lemma 6.
Every I h -free chordal graph G admits a clique cover of size at most h − 1.
Proof. We prove the lemma by induction on h. For h = 2, G itself is a clique and the claim is trivial. Suppose inductively that any I h−1 -free chordal graph admits a clique cover of size at most h − 2, let G be an I h -free chordal graph, and let v be a simplicial vertex of G.
admits a clique cover of size at most h − 2. These h − 2 cliques together with N [v] define a clique cover of G of size at most h − 1.

Lemma 7. Let h ≥ 2 be an integer, let G be a graph, let D be a tree decomposition of G, and let S be any solution for
Proof. Let D be a tree decomposition of G, let S be a solution for I h -IS-Deletion on G, and let G be the graph obtained from G by adding an edge between any pair of vertices contained in the same bag of D. Note that G is a chordal graph, and that D is also a tree decomposition of G . Since being a chordal graph is a hereditary property, it follows that G \ S is chordal. Since G \ S is I h -free, and the property of being I h -free is closed under edge addition, we have that G \ S is also I h -free. Thus, G \ S is an I h -free chordal graph, and Lemma 6 implies that G \ S admits a clique cover of size at most h − 1. Since any clique in G \ S is also a clique in G , and D is a tree decomposition of G , Lemma 4 implies that every clique of G \ S is contained in some bag of D, and therefore there are at most Recall that I h -IS-Deletion is NP-hard even for h = 2, thus the problem cannot be solved in time n f (h) for any function f , unless P = NP.
where n and t are the number of vertices and the treewidth of the input graph, respectively.
Proof. For h = 1 the problem can be trivially solved in linear time, so assume h ≥ 2. Let D be a tree decomposition of G with width t, and let S be an (unknown) optimal solution for I h -IS-Deletion on G. By Lemma 7, there are at most h − 1 bags Since we may assume that D has O(n) nodes [24], we can enumerate the candidate sets of bags by computing a maximum independent set of G[S] using dynamic programming based on treewidth [12] (note that having treewidth at most t is a hereditary property). Note that, by Lemma 7, there exists some of the considered setsS such that V (G) \S = S, and therefore an optimal solution S of I h -IS-Deletion on G can be found We would like to mention that the approach used in the algorithm of Theorem 8 does not seem to be easily applicable to the colorful version of the problem. Indeed, the colored version of Lemma 6 fails: removing a clique from a σ-I h -free chordal graph does not necessarily yield a σ-I h−1 -free chordal graph, and the inductive argument does not apply.
To conclude this section, note that, for any graph H and any instance (G, σ) of Colorful H-IS-Deletion, any edge between two vertices u, v with σ(u) = σ(v) can be safely deleted without affecting the instance. Hence, if H = K 2 we can assume that the input graph is bipartite, and therefore the Colorful K 2 -IS-Deletion problem (where the goal is to hit all edges) is equivalent to computing a minimum vertex cover in a bipartite graph, which can be done in polynomial time. Similarly, the Colorful I 2 -IS-Deletion problem can also be solved in polynomial time, by computing a minimum vertex cover in the bipartite complement of the input graph. This is in sharp contrast to the uncolored version, where both problems are NP-hard [25].

Lower bounds for H-IS-Deletion
In this section we present lower bounds for the H-IS-Deletion problem. Our reductions will be from the 3-Sat problem restricted to clean formulas (see Section 2 for the definition), and are strongly inspired by a reduction of Cygan et al. [13,Theorem 4] for the H-S-Deletion problem when H is the graph obtained from K 2,h by attaching a triangle to each of the two vertices of degree h. More precisely, the reduction from the 3-Sat problem restricted to clean formulas of Cygan et al. [13,Theorem 4] is based on a frame graph that is a simplified version of the general one that we define below, but that enjoys its essential properties, namely that each occurrence of the forbidden (induced) graph H corresponds to a satisfying variable/clause pair. Our technical contribution is to enhance this basic frame graph in order to deal with different forbidden subgraphs H, in particular (as discussed in detail below) by adding edges inside the central part, redefining the "attached graphs" L, changing the adjacencies given by the functions f C, defined below, or adding a new vertex set T into the central part (cf. Figure 1).
We start by presenting the general frame of the reductions together with some generic properties that our eventual instances of H-IS-Deletion will satisfy, which allow to prove in a unified way (cf. Lemma 9) the equivalence of the instances. Variations of this general frame will yield the concrete reductions for distinct graphs H (cf. Theorems 10,11,13,and 14).
General frame of the reductions. Given a clean 3-Sat formula ϕ with n variables and m clauses, we proceed to build a so-called frame graph F H,ϕ . For each graph H considered in the reductions, F H,ϕ will be enhanced with additional vertices and edges, obtaining a graph G H,ϕ that will be the constructed instance of the H-IS-Deletion problem.
Let h be an integer depending on H, to be specified in each particular reduction, let s be the smallest positive integer such that s h ≥ 3n, and note that s Let L be a graph, to be specified according to each particular considered graph H. By attaching a copy of L between two vertices u, v ∈ V (F H,ϕ ) we mean adding a new copy of L, choosing two arbitrary distinct vertices of L, and identifying them with u and v, respectively.
For each variable x of ϕ and for each clause C containing x in a literal ∈ {x,x}, we add to F ϕ a new vertex a x,C, . We also introduce another "dummy" vertex a x . Since ϕ is clean, we have introduced four vertices in F H,ϕ for each variable x. Let a x,C1, , a x,C2,¯ , a x,C3, , a x be the four introduced vertices (recall that x appears at least once positively and negatively in ϕ). We attach a copy of L between the following four pairs of vertices: (a x,C1, , a x,C2,¯ ), (a x,C2,¯ , a x,C3, ), (a x,C3, , a x ), and (a x , a x,C1, ). We denote by A the union of all the vertices in these variable gadgets.
For each clause C of ϕ and for each literal in C, we add to F ϕ a new vertex b C, . Since ϕ is clean, we have introduced two or three vertices in F H,ϕ for each clause C. We attach a copy of L between every pair of these vertices. We denote by B the union of all the vertices in these clause gadgets. This concludes the construction of the frame F H,ϕ ; cf. Figure 1. In all our reductions, the graph G H,ϕ will satisfy the following property: Note that property P3 states that the occurrences of H described above are "representative" of all occurrences of H, in the sense that it is enough that the set X hits these particular occurrences in order to guarantee that G H,ϕ \ X contains no occurrence of H at all. We now show that the above three properties are enough to construct the desired reductions. Suppose first that α is an assignment of the variables that satisfies all the clauses in ϕ, and we define a set X ⊆ V (G H,ϕ ) as follows. For each variable x, add to X all vertices a x,C, such that α( ) is true. If only one vertex was added in the previous step, add to X vertex a x as well. For each clause X, choose a literal that satisfies C, and add to X the set {b C, | = }. By construction we have that |X| = k, and property P3 guarantees that H does not occur in G H,ϕ \ X as an induced subgraph.
Conversely, suppose that there exists X ⊆ V (G H,ϕ ) with |X| ≤ k such that G H,ϕ \ X does not contain H as an induced subgraph. By property P2 and the choice of k, X contains exactly one of (a x,C1, , a x,C3, ) and (a x , a x,C2,¯ ) for each variable x, and exactly |C|−1 vertices in {b C, | ∈ C} for each clause C. We define the following assignment α of the variables: for each variable x, let ∈ {x,x} such that a x,C, ∈ X for some clause C. Then we set α(x) to true if = x, and to false if =x. By the above discussion, this is a valid assignment. Consider a clause C of ϕ, and let be the literal in C such that b C, / ∈ X. Property P3 and the hypothesis that X is a solution imply that there exists a variable x ∈ { ,¯ } such that a x,C, ∈ X, as otherwise there would be an occurrence of H in G H,ϕ \ X. By the definition of α, necessarily α( ) is true, and therefore α satisfies C. Since this argument holds for every clause, we conclude that ϕ is satisfiable.
We now proceed to describe concrete reductions for several instantiations of H. In order to add edges between the enhanced central part M and the sets A, B, we use the following nice trick introduced in [13]. To each pair (C, ), where C is a clause of ϕ and is a literal in C, we assign a function f C, : [h] → [s]. Note that there are s h many such functions, and recall that s has been chosen so that s h ≥ 3n. We assign these functions in such a way that f C, = f C , whenever (C, ) = (C , ); note that this is possible by the choice of s and the fact that, since ϕ is clean, each clause contains at most three literals. We assume henceforth that these functions are fixed.
We start with the following result that provides a tight lower bound for a graph that is very "close" to a clique, namely a clique minus one edge.

Theorem 10. For any integer h ≥ 1, the (K h+2 −e)-IS-Deletion problem cannot be solved in time O * (2 o(t h ) ) unless the ETH fails, where t is the width of a given tree decomposition of the input graph.
Proof. We first treat the case h = 1 separately, by presenting a polynomial-time reduction from the Vertex Cover problem, which is well-known not to be solvable, assuming the ETH, in time 2 o(n+m) on graphs with n vertices and m edges [21,22]. In fact, we will prove the stronger lower bound of O * (2 o(n) ), which implies the lower bound O * (2 o(t) ) corresponding to the case h = 1 claimed in the statement of the theorem. Note that, in the case h = 1, K 3 − e = P 3 . Given an instance G of Vertex Cover, let G be obtained from G by attaching a private neighbor to every vertex of G, and note that |V (G )| = 2|V (G)|. It can be easily verified that the size of a minimum vertex cover of G equals the minimum size of a vertex set of G intersecting all its induced P 3 's. Hence, the (K 3 − e)-IS-Deletion problem cannot be solved in time O * (2 o(n) ) under the ETH.
Suppose henceforth that h ≥ 2, and let H = K h +2 − e for some h ≥ 2 (we relabel h as h in H to keep the index h for the number of columns in the frame graph F H,ϕ ). We will present a reduction from the 3-Sat problem restricted to clean formulas. Given such a formula ϕ, let F H,ϕ be the frame graph described above the statement of the theorem, with h = h , L = K h+2 − e, and T = ∅. In this construction, when attaching copies of L, we choose the attachment vertices to be two distinct vertices of L different from the endvertices of its unique non-edge. Note that this is always possible as h ≥ 2. We proceed to build, starting from F H,ϕ , an instance G H,ϕ of H-IS-Deletion with budget k = 5n − m satisfying properties P1, P2, and P3, and then Lemma 9 will imply the claimed lower bound.
We add an edge between any two vertices w i,j , w i ,j ∈ M with j = j . That is, we turn G H,ϕ [M ] into a complete h-partite graph, where each part has size s. For each clause C and each literal in C, where ∈ {x,x} for some variable x, we add the edges {a C,x, , w f C, (j),j } and {b C, , w f C, (j),j } for every j ∈ [h]. That is, the function f C, indicates the unique neighbor of a C,x, and b C, in the j-th column of M , for every j ∈ [h]. This concludes the construction of G H,ϕ , which clearly satisfies property P1. By the choice of k and the fact that there is a copy of H between the corresponding vertices of A and B (cf. Figure 1), property P2 holds as well. Let X ⊆ V (G H,ϕ ) be a set as in property P3, and letH be an induced subgraph of In the following result we provide an almost tight lower bound for another graph H that is also "close" to a clique, in this case a clique of size h plus two isolated vertices. Since this graph is somehow symmetric to the one considered in Theorem 10, the natural approach is to reverse the roles of neighbors and non-neighbors given by the functions f C, . However, in this way there would be many cliques of size h consisting of a vertex in A ∪ B together with h − 1 of its neighbors in M , which would create many undesired induced occurrences of H with any two vertices anticomplete to such a clique. We circumvent this problem by "reducing" the number of columns of the central part to h − 1, and adding a vertex s 0 to the set T that is complete to M and anticomplete to A ∪ B. This vertex guarantees property P3, at the price of achieving only a near-optimal lower bound 3 for H = K h + I 2 , except for the case h = 1, in which the lower bound is optimal under the ETH. For technical reasons discussed in the proof of Theorem 11, in our construction we need to assume that h ≥ 4. Proof. We first treat the cases h ∈ {1, 2, 3} separately. As in the proof of Theorem 10, we present polynomial-time reductions from the Vertex Cover problem. Again, we will prove the stronger lower bound of O * (2 o(n) ), which implies the lower bound O * (2 o(t) ) corresponding to the cases where h ≤ 3 claimed in the statement of the theorem.
Let first h = 1, and note that K 1 + I 2 = I 3 . We will show that, under the ETH, K 3 -IS-Deletion cannot be solved in time O * (2 o(n) ), which implies, by taking the complement of the input graph, that I 3 -IS-Deletion cannot be solved in time O * (2 o(n) ), concluding the proof. Given an instance G of Vertex Cover, let G be obtained from G by adding, for each edge {u, v} ∈ E(G), a new vertex w and two edges {u, w} and {v, w}. Note that |V (G )| = |V (G)| + |E(G)|, and recall that Vertex Cover cannot be solved in time 2 o(n+m) under the ETH. It can be easily verified that the size of a minimum vertex cover of G equals the minimum size of a vertex set of G intersecting all its K 3 's. Hence, the K 3 -IS-Deletion problem cannot be solved in time O * (2 o(n) ) under the ETH.
Let h = 2. Given an instance G of Vertex Cover, let G be obtained from G by adding |V (G)| isolated vertices. It can be easily verified that the size of a minimum vertex cover of G equals the minimum size of a vertex set of G intersecting all the induced occurrences of K 2 + I 2 .
Finally, let h = 3. Given an instance G of Vertex Cover, let G be obtained from G by adding |V (G)| isolated vertices and, for each edge {u, v} ∈ E(G), a new vertex w and two edges {u, w} and {v, w}. It can be easily verified that the size of a minimum vertex cover of G equals the minimum size of a vertex set of G intersecting all the induced occurrences of Suppose henceforth that h ≥ 4, and let H = K h + I 2 for some h ≥ 4 (again, we relabel h as h in H to keep the index h for the number of columns in the frame graph F H,ϕ ). We will present a reduction from the 3-Sat problem restricted to clean formulas. Given such a formula ϕ, let F H,ϕ be the frame graph described above the statement of the theorem, with h = h − 1, L = K h , and T = {s 0 } where s 0 is a new vertex. We proceed to build, starting from F H,ϕ , an instance G H,ϕ of H-IS-Deletion with budget k = 5n − m satisfying properties P1, P2, and P3, and then Lemma 9 will imply the claimed lower bound.
We add an edge between any two vertices w i,j , w i ,j ∈ M with j = j . That is, we turn , we make v j complete to the j-th column of M (note that the last column of M is not used). We add these edges for two reasons. The first one is to prevent that the non-attachment vertices of L may play the role of the two isolated vertices in a potential occurrence of K h + I 2 . The second one is to prevent that the non-attachment vertices of L may participate in a clique of size h in an occurrence of K h + I 2 . Here is where the hypothesis that h ≥ 4 is important. Indeed, since h ≥ 4, each copy of L contains at least two non-attachment vertices, and the fact that each such vertex is adjacent to a distinct column of M implies that, together with one of the "surviving" attachment vertices and some vertices of M , these non-attachment vertices cannot participate in a clique of size h (for this, we also use that each column of M induces an independent set). This concludes the construction of G H,ϕ , which clearly satisfies property P1, since the vertices in the variable and clauses gadgets have neighbors only in M and within those gadgets.
Let us now argue that G H,ϕ satisfies property P2. Assume for contradiction that there exists a hitting set X of size at most k and a variable x such that none of the pairs (a x,C1, , a x,C3, ) and (a x , a x,C2,¯ ) is entirely in X. The choice of k and the construction of the frame graph F H,ϕ imply that, in that case, there exists an entire copy of L = K h in G H,ϕ \ X. Since the vertices in the variable or clause gadget where this copy of L lies do not have neighbors in other variable or clause gadgets, in order for a copy of H not to occur in G H,ϕ \ X, necessarily X must contain all vertices in all variable and clause gadgets except possibly two of them (the one containing L, and another one that may allow for the occurrence of K h + I 1 ), which clearly exceeds the budget k. An analogous argument applies if we assume that |X ∩ {b C, | ∈ C}| ≤ |C| − 2 for some clause C. Thus, G H,ϕ satisfies property P2.
Finally, let X be a set as in property P3, letH be an induced occurrence of H in G H,ϕ \ X, and let K be the subgraph ofH isomorphic to K h . The choice of X and the discussion above about the edges between M and the copies of L imply that K cannot contain a non-attachment vertex of a copy of L. However, K may contain a variable or clause vertex v ∈ A ∪ B together with h of its neighbors in M , one in each column. Note that, by construction of G H,ϕ , |V (K) ∩ (A ∪ B)| ≤ 1. In order to complete K into K h + I 2 ,H should contain two non-adjacent vertices in A ∪ B that are anticomplete to K. The construction of G H,ϕ forces that these two vertices must be a x,C, ∈ A and b C , ∈ B with (C, ) = (C , ). We distinguish two cases. If |V Proof. We will use the fact that, as observed in [13], K h -S-Deletion can be solved in time O * (2 O(t) ) for every h ≥ 1. This proves the result for = 0, so let now ≥ 1. Without loss of generality, assume that n ≥ h + , as otherwise the solution is zero. Given an n-vertex input graph G together with a tree decomposition of width t, we first solve K h -S-Deletion on G in time O * (2 O(t) ). Let X be a smallest K h -hitting set in G and let |X| = k. Notice that if n ≥ h + , whenever G contains a clique of size h then it contains K h + I as a subgraph as well. Thus, if k ≥ n − (h + ) + 1, then |V (G) \ X| < h + , so we can safely output n − (h + ) + 1 as the size of a smallest (K h + I )-hitting set in G. Otherwise, if k ≤ n − (h + ), we output k. The algorithm is correct by the fact that a (K h + I )-hitting set is not smaller than a K h -hitting set.
By Theorem 2, the lower bound presented in Theorem 10 for H = K h+2 − e is tight under the ETH, and the one presented in Theorem 11 for H = K h + I 2 is almost tight. These two graphs are very symmetric, in the sense that each of them contains two non-adjacent vertices that are either complete or anticomplete to a "central" clique K h (cf. Figure 2). Unfortunately, for graphs without two such non-adjacent symmetric vertices, our framework described above is not capable of obtaining (almost) tight lower bounds. For instance, let H = K h+1 + I 1 , that is, a clique of size h + 1 plus an isolated vertex. Let a ∈ V (H) be any vertex in K h+1 and let b be the isolated vertex. The natural idea in order to obtain a tight lower bound of O * (2 o(t h ) ) would be, starting from the frame graph F H,ϕ described above, to make the vertices a x,C, ∈ A play the role of a, and vertices b C, ∈ B that of b. Then, the functions f C, would indicate, for each vertex a x,C, (resp. b C, ), its unique neighbor (resp. unique non-neighbor) in each of the columns of M . However, this idea does not work for the following reason: many undesired copies of H appear by interchanging the expected roles of vertices a x,C, and b C, , and selecting, in each column of M , any of the s − 1 non-neighbors of a and any of the s − 1 neighbors of b. We overcome this problem by "pledging" one column of M and introducing a sentinel vertex s 0 ∈ T that is complete to M and the vertices a x,C, , and anticomplete to the vertices b C, . This vertex "fixes" the roles of vertices in A and B at the price of losing one column of M , hence getting (by Lemma 9) a weaker lower bound of O * (2 o(t h−1 ) ), as in Theorem 11. In the following theorem we formalize this idea and we extend it to a more general graph H, namely K h+1 + v x for 0 ≤ x ≤ h − 1, defined as the graph obtained from K h+1 by adding a vertex v adjacent to x vertices in the clique (cf. Figure 2). We will need an extra sentinel vertex in T for each of the neighbors of v in the clique, losing one column of the central part M for each of them.

Theorem 13.
Let h ≥ 1 and 0 ≤ x ≤ h − 1 be integers and let K h+1 + v x be the graph obtained from K h+1 by adding a vertex adjacent to x vertices in the clique. Then, unless the ETH fails, the where t is the width of a given tree decomposition of the input graph.
Proof. Let H = K h +1 + v x for h ≥ 1 and 0 ≤ x ≤ h − 1 (again, we relabel h as h in H to keep the index h for the number of columns in the frame graph F H,ϕ ). We will again present a reduction from the 3-Sat problem restricted to clean formulas. Given such a formula ϕ, let F H,ϕ be the frame graph described above with h = h − x − 1, T = {s 0 , s 1 , . . . , s x }, and L = K h +1 if x = 0 and L = H if x ≥ 1. We proceed to build, starting from F H,ϕ , an instance G H,ϕ of H-IS-Deletion with budget k = 5n − m satisfying properties P1, P2, and P3.
We first add an edge between any two vertices w i,j , w i ,j ∈ M with j = j . That is, we turn G H,ϕ [M ] into a complete h-partite graph, where each part has size s. We make Moreover, in the case x = 0 (that is, when H = K h +1 + I 1 , which is disconnected), for each copy of L in G H,ϕ [B], we do the following. Let v 1 , . . . , v h −1 be the vertices of L distinct from the two attachment vertices, ordered arbitrarily. For j ∈ [h − 1], we make v j complete to the j-th column of M . We add these edges to prevent that the non-attachment vertices of L may play the role of the isolated vertex in a potential occurrence of K h +1 + I 1 .
This concludes the construction of G H,ϕ , which can be easily seen to satisfy properties P1 and P2 using arguments analogous to those used in the proof of Theorem 10. Let X ⊆ V (G H,ϕ ) be a set as in property P3, and letH be an induced subgraph of G H,ϕ \ X isomorphic to H. Finally, when x = 0, the same arguments yield that V (H) ∩ M induces a clique K of size h . Note that K must contain one vertex from each column of M and the whole set T . By construction of G H,ϕ , the only vertices in A ∪ B that can be complete to K (resp. complete to T \ {s 0 } and anticomplete to K \ (T \ {s 0 })) are the vertices a x,C, (resp. b C, ). Hence, the two vertices in V (H) ∩ (A ∪ B) must be a x,C, ∈ A and b C , ∈ B with (C, ) = (C , ), and therefore property P3 also follows and we are done again by Lemma 9.
It is worth mentioning that the lower bound given in Theorem 13 can be strengthened to O * (2 o(t h−x−1 ) ), wherex = min{x, h − 1 − x}, by using the following trick. If the vertex not belonging to the clique K h+1 (vertex b in Figure 2) has more than h−1 2 neighbors in the clique (i.e., ifx = h − 1 − x), we can interchange, for vertices b C, ∈ B, the roles of neighbors/non-neighbors of the set T \ {s 0 } and the vertices in M given by the functions f C, . Doing this modification, an analogous proof works and we can keep the number of columns of M to be h −x − 1, which is always at least h−3 2 . We omit the details. Another direction for transferring the lower bounds of Theorem 10 and Theorem 13 to other graphs H is to consider complete bipartite graphs. Basically, property P3 states that all the copies of the former set A behave in a similar way. With this in mind, it is easy to see that Lemma 9 still holds if we replace, for an instance G H,ϕ constructed in this reduction, property P3 by property P3 .
Consider a set X ⊆ V (G H,ϕ ) as in property P3 , and letH be an induced subgraph of G H,ϕ \ X isomorphic to K h,h . By construction of G H,ϕ , one of the two parts ofH must lie entirely inside M , as there are no edges among distinct variable or clause gadgets. Since the other part ofH must lie entirely inside A ∪ B, the choice of the functions f C, implies that the only vertex sets of size h in A ∪ B that are complete to a set of h non-adjacent vertices are of the form {a β x,C, | β ∈ [h − 1]} ∪ {b C, } for some clause C and a literal ∈ {x,x}, hence property P3 holds and the theorem follows.
Note that the proof of Theorem 14 works for both K h,h -IS-Deletion and K h,h -S-Deletion, since all the occurrences of K h,h in the constructed graph G H,ϕ are induced. Hence, as the particular case of Theorem 14 for h = 2 we get the following corollary, which answers a question of Mi. Pilipczuk [28] about the asymptotic complexity of C 4 -S-Deletion parameterized by treewidth.

Corollary 15.
Neither C 4 -IS-Deletion nor C 4 -S-Deletion can be solved in time O * (2 o(t 2 ) ) unless the ETH fails, where t is the width of a given tree decomposition of the input graph.
As mentioned in [28], C 4 -S-Deletion can be easily solved in time O * (2 O(t 2 ) ). This fact together with Theorem 2 imply that both lower bounds of Corollary 15 are tight.
We can obtain lower bounds for other graphs H that are "close" to a complete bipartite graph. Indeed, note that the lower bound of Theorem 14 also applies to the graph H obtained from K h,h by turning one of the two parts into a clique: the same reduction works similarly, and the only change in the construction is to turn the whole central part M into a clique. We can also consider complete bipartite graphs K a,b with parts of different sizes, by letting the number of columns of the central part M be equal to max{a, b}, hence obtaining a lower bound of O * (2 o(t max{a,b} ) ). Similarly, we can also turn one of the two parts of K a,b into a clique, and obtain the same lower bound. In particular, in this way we can obtain a lower bound of O * (2 o(t h ) ) for the graph H obtained from K h+3 by removing the edges in a triangle.

Lower bounds for Colorful H-IS-Deletion
Our main reduction for the colored version is again strongly inspired by the corresponding reduction of Cygan et al. [13,Theorem 2] for the non-induced version, again based on a reduction from the 3-Sat problem restricted to clean formulas and the frame graph defined in Section 4. The main difference with respect to their reduction is that in the non-induced version, the graph H is required to contain a connected component that is neither a clique nor a path, while for the induced version we only require a component that is not a clique, and therefore we need extra arguments to deal with the case where all the connected components of H are paths. In particular, in the proof of Theorem 16, the definition of the graphs L A and L B where the graph H 0 is a path is inspired from the proof of [13,Theorem 22].

Theorem 16. Let H be a graph having a connected component on h vertices that is not a clique. Then Colorful H-IS-Deletion cannot be solved in time
) unless the ETH fails, where t is the width of a given tree decomposition of the input graph. H 0 , H 1 , . . . , H p be the connected components of H, where H 0 is not a clique. Hence, |V (H 0 )| ≥ 3. As in Section 4, we will again reduce from the 3-Sat problem restricted to clean formulas. Given such a formula ϕ with n variables and m clauses, we proceed to construct an instance (G H,ϕ , σ) of Colorful H-IS-Deletion such that ϕ is satisfiable if and only if G has a set X ⊆ V (G) of size at most k := 15n − 4m hitting all induced σ-H-subgraphs of G, satisfying properties P1, P2, and P3 (the latter one, concerning only colorful copies of H, of course), and then Lemma 9 will imply the claimed lower bound. The choice of the budget of the current reduction will become clear below, and does not affect the main properties of the reduction.

Proof. Let
We start with the frame graph F H,ϕ defined in Section 4, with h = |V (H 0 )| − 2 ≥ 1, T = ∅, and L to be specified later. Let the vertices of H 0 be labeled z 0 , z 1 , . . . , z h+1 such that {z 0 , z h+1 } / ∈ E(H 0 ); note that this is always possible since H 0 is not a clique. We define the H-coloring σ of G H,ϕ starting from the vertices of F H,ϕ except for the nonattachment vertices of the graphs L, whose coloring will be defined later together with the description of L. Namely, for each variable x and each clause C containing ∈ {x,x}, we define σ(a x,C, ) = σ(a x ) = z 0 and σ(b C, ) = z h+1 . For every i ∈ [s] and j ∈ [h] we define σ(w i,j ) = z j . That is, the vertices a x,C, (resp. b C, ) are mapped to z 0 (resp. z h+1 ), and the whole j-th column of M is mapped to z j for j ∈ [h]. We now add edges among the already colored vertices as follows. Within M , the edges mimic those in H 0 : for any two vertices w i,j , w i ,j ∈ M we add the edge {w i,j , w i ,j } in G H,ϕ if and only if {z j , z j } ∈ E(H 0 ). As for the edges between A ∪ B and M , the functions f C, capture the existence or non-existence of the edges in H 0 between the corresponding vertices. Namely, for each clause C and each literal in C, where ∈ {x,x} for some variable x, and for every j ∈ [h], we do the following: We now proceed to describe the graph L together with its H-coloring. In fact, we define different (but very similar) graphs L for the copies to be attached in A and B; we call them L A and L B , respectively. We start with the definition of L A , and we distinguish two cases according to H 0 : Suppose first that H 0 is not a path. As observed in [13], it is easy to see that every graph that is not a path contains at least three vertices that are not separators. Let z β , z γ be two such non-separating vertices of H 0 different from z 0 . We define L A as the graph obtained from the disjoint union of three copies In both the above cases, when we attach a copy of L A between two vertices u, v ∈ A, we identify u and v with the vertices z 0 of the first and third copies of H 0 , respectively. The H-coloring σ of L A is defined naturally, that is, each vertex gets its original color in H 0 .
The graph L B is defined in a completely symmetric way, just by replacing vertex z 0 ∈ V (H 0 ) in the above definition of L A by vertex z h+1 ∈ V (H 0 ). Claim 17. In the graph L A (resp. L B ) defined above, where u and v are the attachment vertices, there are exactly two vertex sets X 1 , X 2 ⊆ V (L A ) (resp. V (L B )) of minimum size hitting all induced σ-H 0 -subgraphs of L A (resp. L B ), |X 1 | = |X 2 | = 2, X 1 ∩ {u, v} = {u}, and X 2 ∩ {u, v} = {v}.
Proof. By symmetry, it suffices to present the proof for L A . In order to prove the claim, it is enough to prove that there are exactly three induced σ-H 0 -subgraphs in L A , corresponding to the three copies H 1 0 , H 2 0 , H 3 0 of H 0 . Indeed, once this is proved, there are exactly two

minimum-sized hitting sets in L
. (For notational simplicity, we interpret an induced σ-H 0 -subgraph as an induced subgraph of L A isomorphic to H 0 , with matching colors.) We distinguish two cases depending on H 0 .
If H 0 is not a path, the existence of such anH 0 in L A would imply, by the construction of L A and since H 0 is connected, that at least one of z β and z γ is a separator in H 0 , contradicting their choice.
Otherwise, if H 0 is a path, first note that since the vertex . Since in L A , vertex z γ was chosen as an internal vertex of H 0 , let z i and z j be the two neighbors of z γ inH 0 . Recall that in L A we added all edges between V (H 2 0 ) \ {z γ } and V (H 3 0 ) \ {z γ }. We distinguish two cases: Suppose first that both z i , z j ∈ V (H 2 0 ) \ V (H 3 0 ) or z i , z j ∈ V (H 3 0 ) \ V (H 2 0 ). Assume that the former case holds -the other one being symmetric-and let z r be a vertex ofH 0 in V (H 3 0 ) \ V (H 2 0 ), which exists by hypothesis. Then {z γ , z i , z j , z r } induces a C 4 inH 0 , a contradiction sinceH 0 is a path. Otherwise, suppose without loss of generality that z i ∈ V (H 2 0 ) ∩ V (H 3 0 ) and z j ∈ V (H 3 0 ) ∩ V (H 2 0 ). Then {z γ , z i , z j } induces a C 3 inH 0 (for example, in Figure 3(b), the vertices {z 1 , z 0 , z 2 } with z 1 ∈ V (H 2 0 ) ∩ V (H 3 0 ) induce a C 3 ), a contradiction again.
Note that Claim 17 justifies the budget k of our eventual instance (G H,ϕ , σ) of Colorful H-IS-Deletion: any optimal solution X needs to contain one of the pairs (a x,C1, , a x,C3, ) and (a x , a x,C2,¯ ) for each variable x, and |C| − 1 vertices in {b C, | ∈ C} for each clause C.
Finally, for every i ∈ [p], add k + 1 disjoint copies of H i , and color their vertices according to their colors in H. This concludes the construction of (G H,ϕ , σ), which clearly satisfies property P1. Note that since for every i ∈ [p] the number of copies of H i in G H,ϕ exceeds the budget, hitting all colorful induced copies of H in G H,ϕ with at most k vertices is equivalent to hitting all colorful induced copies of component H 0 . Therefore, Claim 17 implies that (G H,ϕ , σ) satisfies property P2 as well.
Consider a set X ⊆ V (G H,ϕ ) as in property P3, and letH be an induced σ-H-subgraph of G H,ϕ \ X. Since for every j ∈ [h] the only vertices of G H,ϕ colored z j by σ are those in the j-th column of M , we conclude that, for every j ∈ [h],H contains exactly one vertex from the j-column of M . Since H 0 is connected and the only vertices in G H,ϕ colored z 0 (resp. z h+1 ) with neighbors in M are those of type a x,C, (resp. b C, ), it follows that H contains exactly one vertex a x,C, and exactly one vertex b C , . The properties of the functions f C, , which define the edges between A ∪ B and M , and the fact thatH needs to be an induced σ-H-subgraph, imply that necessarily (C, ) = (C , ), and therefore (G H,ϕ , σ) satisfies property P3 and the theorem follows by Lemma 9.
When H is a connected graph, the lower bound of Theorem 16 together with the algorithms given by Proposition 5 and Theorem 3 completely settle, under the ETH, the asymptotic complexity of Colorful H-IS-Deletion parameterized by treewidth. Note that, in particular, Theorem 16 applies when H is path, in contrast to the subgraph version that can be solved in polynomial time [13].
Therefore, what remains is to obtain tight lower bounds when H is disconnected. In particular, Theorem 16 cannot be applied at all when all the connected components of H are cliques, since the machinery that we developed (inspired by Cygan et al. [13]) using the framework graph F H,ϕ crucially needs two non-adjacent vertices in the same connected component. Let us now focus on those graphs, sometimes called cluster graphs in the literature.
As mentioned in Section 3, both Colorful K 2 -IS-Deletion and Colorful I 2 -IS-Deletion can be solved in polynomial time. In our next result we show that if H is slightly larger than these two graphs (namely, K 2 or I 2 ), then Colorful H-IS-Deletion becomes hard. Namely, we provide a single-exponential lower bound for the following three graphs H on three vertices that are not covered by Theorem 16: K 3 , I 3 , and K 2 + K 1 . Note that these lower bounds are tight by the algorithm of Theorem 3.

Theorem 18.
Let H ∈ {K 3 , I 3 , K 2 + K 1 }. Then, unless the ETH fails, the Colorful H-IS-Deletion problem cannot be solved in time O * (2 o(t) ), where t is the width of a given tree decomposition of the input graph.
Proof. We will prove that none of the considered problems can be solved in time 2 o(n) under the ETH, which clearly implies the statement of the theorem. For this, we reduce from the Vertex Cover problem restricted to input graphs with maximum degree at most three. To see that this problem cannot be solved in time 2 o(n) under the ETH, where n is the number of vertices of the input graph, one can apply the classical NP-hardness reduction [20] from 3-Sat to Vertex Cover, but restricting the input formulas to be clean. Then the result follows from Lemma 1.
We first present a reduction for H = K 3 . Given an instance G of Vertex Cover, with |V (G)| = n, |E(G)| = m, and ∆(G) ≤ 3, we proceed to construct an instance (G K3 , σ) of H-IS-Deletion with |V (G K3 )| = O(n) such that G has a vertex cover of size at most k if and only if G K3 has a set of size at most k + m hitting all (induced) σ-K 3 -subgraphs. Note that this will prove the desired result, as tw(G K3 ) ≤ |V (G K3 )| = O(n), and a tree decomposition of G K3 achieving that width consists of just one bag containing all vertices.
for cliques and independent sets are isolated exceptions. The reason is that, in contrast to the subgraph version, when hitting induced subgraphs, edges and non-edges behave essentially in the same way when performing dynamic programming, as one has to keep track of both the existence and the non-existence of edges in order to construct the tables, and storing this information seems to be unavoidable.
Concerning the algorithm for I h -IS-Deletion running in time 2 O(t) · n h (Theorem 8), it would be interesting to find, if it exists, an FPT algorithm parameterized by both t and h while keeping the dependency on t single-exponential, maybe even being linear in n.
As for Colorful H-IS-Deletion, in view of Theorems 3, 5, 16, and 18, only the cases where H is a disjoint union of at least two cliques and |V (H)| ≥ 4 remain open. In particular, we do not know the tight function when H is an independent set or a matching with |V (H)| ≥ 4.