Connected search for a lazy robber

The node‐search game against a lazy (or, respectively, agile) invisible robber has been introduced as a search‐game analogue of the treewidth parameter (and, respectively, pathwidth). In the connected variants of the above games, we additionally demand that, at each moment of the search, the clean territories are connected. The connected search game against an agile and invisible robber has been extensively examined. The monotone variant (where we demand that the clean territories are progressively increasing) of this game corresponds to the graph parameter of connected pathwidth. It is known that the price of connectivity to search for an agile robber is bounded by 2, that is, the connected pathwidth of a graph is at most twice (plus some constant) its pathwidth. We investigate the study of the connected search game against a lazy robber. A lazy robber moves only when the cops' strategy threatens the vertex that he currently occupies. We introduce two alternative graph‐theoretical formulations of this game, one in terms of connected tree‐decompositions and one in terms of (connected) layouts, leading to the graph parameter of connected treewidth. We observe that the connected treewidth parameter is closed under contractions and prove that for every k≥2 , the set of contraction obstructions of the class of graphs with connected treewidth at most k is infinite. Our main result is a complete characterization of the obstruction set for k=2 . We also show that, in contrast to the agile robber game, the price of connectivity is unbounded.

moment of the search, the clean territories are connected. The connected search game against an agile and invisible robber has been extensively examined. The monotone variant (where we demand that the clean territories are progressively increasing) of this game corresponds to the graph parameter of connected pathwidth. It is known that the price of connectivity to search for an agile robber is bounded by 2, that is, the connected pathwidth of a graph is at most twice (plus some constant) its pathwidth. We investigate the study of the connected search game against a lazy robber. A lazy robber moves only when the cops' strategy threatens the vertex that he currently occupies. We introduce two alternative graphtheoretical formulations of this game, one in terms of connected tree-decompositions and one in terms of (connected) layouts, leading to the graph parameter of connected treewidth. We observe that the connected treewidth parameter is closed under contractions and prove that for every k 2 ≥ , the set of contraction obstructions of the class of graphs with connected treewidth at most k is infinite. Our main result is a complete characterization of the obstruction set for k = 2. We also show that, in contrast to the agile robber game, the price of connectivity is unbounded.

| INTRODUCTION
In a graph search game the competing parts are a group of cops and a robber that move in a graph while having opposite goals. The goal of the cops is to capture the robber, while the robber is trying to avoid capture. Typically, the robber is assumed to be omniscient, implying that he/she always makes the best possible move towards avoiding capture. Numerous and quite diverse variants of this game can be defined, depending on the rules that determine how the cops and the robber can move and what the definition of "capture" is. A search strategy represents a series of moves that eventually yield the capture of the robber. Given the rules of the game, the cost of a search strategy is the maximum number of cops simultaneously present on the graph during the search. Then the corresponding search number of the graph is defined as the minimum cost of a search strategy. The study of graph searching parameters is an active field of graph theory as several important graph parameters have their search-game analogues that provide useful insights. For related surveys, see [2,3,8,24,26,45].
1.1 | Node-search against an invisible searcher by cops. In other words, a monotone strategy should guarantee that the "clean" territory (i.e., the set of vertices from which the robber has been expelled) is gradually increasing (and, certainly, finally occupies the whole graph). The monotone search number is defined as the minimum number of cops required to capture the robber by using a monotone search strategy. We say that a search game is monotone when the corresponding search number is equal to the nonmonotone one, that is, monotone strategies are as good as the nonmonotone ones.

| Agility and laziness
Different variants of the game arise depending on the mobility rules of the robber, for example, a robber can be lazy or agile. A lazy robber residing on vertex v may move only if the next move of the search strategy is a placement of a cop on v. In other words, the lazy robber stays put, unless his position is threatened by the cop's strategy. On the other hand, an agile robber may always move no matter what the next move of the search strategy is. The distinction between a lazy and an agile robber has been introduced for the first time in [16]. Parameterizations of the game that oscillate between the lazy and agile variants have been studied in [47] (see also [28]). There is an extensive amount of research on the four versions of the search game that is generated by the above variations. The reason for this is that they correspond to well-studied graph-theoretical parameters. The monotone search number of a graph G against an agile (resp., lazy) robber is equal to the pathwidth (resp., treewidth) of G plus one [35,43,50,37,38,16]. Also, it was proven that the nonmonotone versions of the above variants are equal to their monotone counterparts [50,9,10,25,40]. This induced a clear landscape on how to connect treewidth and pathwidth to graph searching and established the intuition that the qualitative difference between the two parameters is expressed by the agility or the laziness of the robber. Moreover, the monotonicity proofs were based on min-max theorems and alternative definitions of the corresponding parameters [50,10,25].

| The connectivity issue
All the above variants are assuming that there is some place "somewhere outside" the graph where the cops go when they are removed from the graph and they stay there to readily be placed to some vertex, as required by the search strategy. For this reason, such search games have also been called "helicopter search games" (as suggested in [50]). This is not always realistic, as the search may take place in a building or in a system of caves where the cops start their persecution from some particular vertex (the entry point) and they do not have the ability of "teleporting themselves" to nonneighboring vertices. This natural restriction was considered and studied for the first time in [6]. A search strategy is connected when at each moment of the search the clean territories induce a connected subgraph. 1 This inspired the question on the "price of connectivity," asking whether there is some universal constant c such that the connected search number is no more than c times its nonconnected counterpart. In its original form, this question was made in [6] for the agile variant and, in the same paper, it was answered affirmatively for the case of trees (see also [5,[21][22][23]27,29,44] for related results). Later, it was proved for all graphs by Dariusz Dereniowski in [17]. In particular, in [17], a connected counterpart of pathwidth was suggested, called connected pathwidth, that is equivalent to the monotone-connected agile search number. Then it was proved that this parameter is always upper bounded by twice the pathwidth plus one.
Much less is known about the nonmonotone variants of the connected search game. The monotonicity question for the connected search against an agile robber was resolved negatively in [51]. Analogous negative results have been derived in the case where the fugitive is visible and agile [30] (which is equivalent to the invisible and lazy case).

| Connected treewidth
This paper initiates the study of the monotone-connected search against a lazy robber. Our first step is to provide two alternative definitions of this parameter: one in terms of treedecompositions and one in terms of layouts. Before we proceed, we need to give the formal definition of tree-decompositions and treewidth.
A tree-decomposition of a graph Figure 1 illustrates the difference between connected and nonconnected tree-decompositions. If, in the above definitions, T is a path instead of a tree, then we define the notion of a pathdecomposition and the parameter of pathwidth which we denote by G ( ) pw accordingly. Observe that none of the tree-decompositions of a disconnected graph is connected. So from now on, we only consider connected graphs. The connected treewidth of a (connected) graph is defined over the set of connected tree-decompositions as follows: In the case where T is a path and r is one of its endpoints, we obtain the corresponding notion of a connected path-decomposition and the parameter connected pathwidth which we denote by G ( ) cpw (this is the same as the connected pathwidth given in [17]). Our first result is that connected treewidth is equal to the monotone, connected, and lazy search number minus one. Our proof (see Section 3) comes together with a second equivalent definition, given in terms of layouts. A layout is connected if every prefix induces a connected subgraph. If we apply the standard layout-based definition of treewidth, given in [16] (see also [14]) to connected layouts, then we again obtain the monotone, connected, and lazy search number minus one. We stress that both equivalences constitute natural adaptations of known definitions of treewidth to the connected setting. They also provide a useful combinatorial background for further investigations.

| Alternative notions of connected treewidth
We now make a short deviation as this is not the first time a "connected" counterpart of treewidth is proposed. We give two alternative definitions below.
• In [29], Fraigniaud and Nisse consider tree-decompositions with the following additional connectivity property: for every edge e of the tree-decomposition T ( , )  , if T 1 and T 2 are two connected components created after removing e from T, then, for every i {1, 2} ∈ , the union of the bags of T i induces a connected subgraph of G. The main result of [29] was that if we define connected treewidth under this condition, then the resulting parameter is again equal to treewidth. Interestingly, this equality breaks if we transfer this definition to pathwidth. To see this, take the graph K 1,3 (2) in Figure 2 and observe that pw while, under the connectivity restriction of [29], its connected pathwidth is 3.
• Another way to define connected treewidth is to consider tree-decompositions where for every t V T ∈ , the bag X t induces a connected subgraph of G. Let us call the variant bagconnected treewidth. This definition was introduced independently, in different contexts, by Jégou and Terrioux in [32] and by Diestel and Müller in [19]. The definition of bag-connected treewidth is quite natural and defines a different parameter that has been the subject of several investigations both in theory and in applications.
The results in [32] used bag-connected tree-decompositions in the context of solving Constraint Satisfaction Problems (CSPs) and they show experimentally that this leads to significant improvements in the resolution of CSPs by decomposition methods. On the other hand, Diestel and Müller [19] initiated a combinatorial study of bag-connected treewidth and revealed interesting relations with graph-geometric parameters, such as the geodesic cycle number and graph hyperbolicity (see also [31]).
It follows from the definitions that, for connected graphs, the connected treewidth is sandwiched between treewidth and bag-connected treewidth. Moreover the three parameters can be different. Observe that the bag-connected treewidth of the graph G of Figure 1 is 4 while as noticed before G G ( ) = 2, ( ) = 3 tw c t w .

| Contraction obstructions
We say that a graph H is a contraction of G, denoted by H G ≼ , if a graph isomorphic to H can be obtained from G by a series of edge contractions (see definition in Section 2.1). We also say that H is a minor of G, denoted by H G ≤ , if H is a contraction of some subgraphs of G. We say that a graph class  is closed under minors (contractions, respectively) if every minor (contraction, respectively) of a graph in  belongs to . We also define the minor obstructions (contraction obstructions, respectively) of , denoted by obs ( )  ≤ (obs ( )  ≼ , respectively), as the set of all minor (contraction, respectively) minimal graphs that do not belong to . It is easy to see that when  is minor (contraction, respectively) closed, then obs ( )  ≤ (obs ( )  ≼ , respectively) provides a complete characterization of a minor-closed class : a graph belongs to  if and only if it excludes all graphs in obs ( )  ≤ (resp., obs ( )  ≼ ) as minors (contractions, respectively). Moreover, in the case of the minor relation, we know from the theorem of Roberston and Seymour [48] (that was the main outcome of the Graph Minors series) that the set obs ( )  ≤ is always finite and therefore the aforementioned characterization provides a finite characterization of any minor-closed class in 2 | FORMAL DEFINITIONS OF THE SEARCH GAME AND ITS VARIANTS

| Numbers and sequences
All numbers in this paper are integers. Given a number n, we define n n [ ] = {1, …, } and, given two numbers n m , where n m ≤ , we define n m n m [ , ] = { , …, }. Given a set U and a collection  of subsets of U , we set We also denote by 2 U the set of all subsets of U . Given a finite set U , a sequence σ over U is a bijection σ U U is the position of x in σ. For two elements x and y U ∈ , we write x y < σ if σ x σ y ( ) < ( ). To simplify the notation we define σ σ i = ( ) We also define the sets σ , and σ i ⩾ are defined similarly and, for i j Alternatively, we also denote a sequence by σ σ σ = ,…, n 1 〈 〉. We define the concatenation of two sequences σ and σ′ over the sets U and U′, respectively, as the sequence σ σ′ ⧹ as the sequence σ′ that is recursively defined as follows:

| Graphs
We consider undirected, simple graphs. Given a graph G, we let V G ( ) and E G ( ) denote its vertex and edge set, respectively. We set G V G = ( ) | | | |. We use the shortcut xy for an edge x y { , } of G, agreeing that xy and yx denote the same edge.
is the set of vertices of G that does not belong to S and are adjacent to some vertex in S. The subgraph induced by S, denoted G S [ ], has vertex set S and edge set yx E x y S is a cut-vertex if x { } is a separator. The set of cut-vertices of a graph G is denoted C G ( ). A graph G is biconnected if it is connected and C G ( ) = ∅. A biconnected component of a graph is any biconnected subgraph of G that is vertexmaximal. A bridge in a graph G is an edge whose removal increases the number of connected components. A separating edge is an edge xy E G ( ) ∈ such that x y { , } is a minimal separator of G. Observe that a bridge is not a separating edge.
Given a tree T and x y V T , ( ) ∈ , we denote by xTy the unique path in T that has endpoints x and y.
Given a graph G, a search strategy on G is a sequence S S = ,…, r 1  〈 〉, with r  ∈ , over the set of subsets of vertices of V G ( ) where , the symmetric differeninduces a connected subgraphce of S i and S i+1 has cardinality one.
Notice that a search strategy  indicates a sequence of moves of cops on G. Such a move may be either a placement of a cop to a vertex or the removal of a cop from a vertex. To see this, consider consecutive elements S i−1 and S i of  for some i , then the corresponding move is the placement of a cop on vertexv.  〈 〉 be a search strategy on G. We define the sequence of robber spaces of an agile robber with respect to  as the sequence there is a path from a vertex u F i−1 ∈ to v whose vertices except u belong to V S − } i .

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| Similarly, we define the sequence of robber spaces of a lazy robber with respect to  as the sequence to v whose vertices except u belong to V S − } i .

| Properties of search strategies
Given the type { , } t a l ∈ of the robber (agile or lazy) and a search strategy  on G, We say that the search strategy  is Observe that connected strategies only exist if the graph is connected. We define If in the above definitions we consider only monotone strategies, then we obtain the parameters mans and mlns. In the context of connected graphs, restricting to connected searches yields to the definition of the , , cans clns mcans, and mclns parameters.

| Search strategies of rooted graphs
A search strategy on a rooted graph G G = ( , )  is defined as a search strategy on G with the difference that S V = ( ) 1  (i.e., the cops first occupy the root vertices). The set of robber space and clean territories ( ( ) ) is defined as in the case of unrooted graphs with the difference that now F V = ( ) 1  . The notions of monotonicity, connectivity, and completeness of a search strategy are defined as in the case of unrooted graphs (where connectivity is interpreted as the connectivity of rooted graph).

| Supporting sets
Let σ be a layout of G on n vertices. For every i n [ ] ∈ , we define the tree-supporting set of position i as ADLER ET AL.
We also define the path-supporting set of position i as Figure 4 for an example). We also define the tree vertex separation number of G and the path vertex separation number of G as respectively. 2 It is known that, for every graph mans ans pw [35,43,10,40,37,38]. In the context of connected graphs, if we restrict in (3) and (4) the set of layouts to be connected (i.e., we replace G ( ) , then the defined parameters are the connected tree vertex separation number of G and the connected path vertex separation number of G denoted G ( ) ctvs and G ( ) cpvs , respectively. All the definitions of this subsection are extended to rooted graphs accordingly to the definition of rooted layout and connectivity of rooted graphs. For this, we replace G by G ( , )  and we define , and G ( , )  cpvs .

| Contractions
Let G be a graph. Contracting an edge e xy E G = ( ) ∈ yields the graph G e ∕ obtained by removing x and y from G, introduce a new vertex and make it adjacent with all vertices in . Notice that contraction of an edge does not create multiple edges and the resulting graph remains simple. If e is incident to a degree-2 vertex x, then contracting e is equivalent to dissolving x, that is removing x from the vertex set and adding an edge between its two neighbors. If F is a subset of edges of G, then G F ∕ is the graph obtained by contracting the edges of F . Observe that the order of the contraction does not matter. We present below an extension of contraction to rooted graphs.
In the literature [35] the path vertex separation number is known as the ≪vertex separation number≫. This alternative term is adopted in this paper to make clear the distinction with the concept of ≪tree separation number≫.
We say that H holds if a rooted graph isomorphic to H ( , )  can be obtained after a series of edge contractions on G, under the constraint that no path between two vertices of V ( )  can be contracted to a single vertex. We say that H ( , )  is a proper For an edge e incident to at least one nonroot vertex, we let e G∕ denote the rooted graph obtained by contracting the edge e. Similarly, if F is a set of edges not containing the edge set of a path between two root vertices, then F G∕ is the rooted graph resulting from the contraction of the results from the contraction of an edge incident to a root vertex of , then v is a root vertex of .
Let k 0 ≥ . Given a set  of k-rooted graphs and a k-rooted graph G, we say that Proof. Let n be the number of vertices of G 2 . Let e xy = be an edge of G 2 such that Let v e be the vertex resulting from the contraction of e and let i σ x = ( ) and j σ y = ( ), where i j < . We construct the following layout σ′ of G′: As contracting an edge does not disconnect a graph, the fact that σ is connected implies that every subgraph induced by a prefix of σ′ is connected and henceforth, σ . To see this, observe that for

| EQUIVALENCE OF PARAMETERS
In this section, we prove the following theorem.
The proof is a consequence of the next three lemmata.

Lemma 2. Let G be a connected graph. Then
Proof. Let n be the number of vertices of G. Let S S = ,…, r 1  〈 〉 be a connected, monotone, and complete strategy against a lazy robber certifying that G k ( ) + 1 mclns ≤ . We associate with  the layout σ of G defined as the ordering in which the vertices are first occupied by a cop. More formally, consider the sequence be the sequence of indices such that j = 1 1 and This implies that every prefix of σ induces a connected subgraph of G, and thereby σ is a connected layout of G.
We now prove that , the tree- Proof. Let n be the number of vertices of G. Suppose without loss of generality that . Consider the graph Ĝ obtained from G by completing, for every that Ĝ i has a treedecomposition T ( , ) i i  of width at most k that is also a connected tree-decomposition of G i . We assume that for some i n T is a tree-decomposition of Ĝ i such that: For the induction base, the four conditions trivially hold for i = 1, by setting T = ({1}, ) 1 ∅ , and σ = {{ }} 1 1  .
(see, e.g., Lemma 4 in [12]). We construct a tree-decomposition T ( , ) Let us now prove that T ( , ) , let P j denote the unique path in T i+1 between nodes 1 and j and set Observe that by the induction hypothesis, if V h induces a connected subgraph of G, then for every subtree T′ of induces a connected subgraph of G. To see this, recall that σ is a connected layout of G and therefore σ i+1 has some neighbor, say x, that belongs to σ i ≤ .
As a connected tree-decomposition T has a root node r such that Let us consider a connected layout σ of the tree We define the following three operations that generate a sequence of sets: where δ is an arbitrary connected layout of G ( , )  .

The proof goes by induction on
, proving that G i has a search strategy 〉 against a lazy robber such that  (3) hold if we replace i by i + 1. For this, notice first that σ i+1 is a leaf of T i+1 and let σ h be its unique neighbor in T i+1 .
Observe that as G i+1 is connected, B i+1 is connected as well. We construct the following search strategy of G i+1 : Figure 5 for a visualization of the sets X X , . The next j 1 moves consist in removing cops from X X σ σ i i+1 ⧹ one by one in an arbitrary order. Observe then that the current set of cops is X X X X ∩ are not currently occupied by a cop, then the next j 2 moves consist in placing them back one by one in an arbitrary order. Observe that, by the laziness of the robber, after each of these j j + 1 2 moves, the clean territory remains unchanged, that is for every , we can complete the search strategy i+1  by adding the cops of X X σ σ i h +1 ⧹ one by one following a connected search strategy of the connected rooted graph B i+1 .
To prove the monotonicity of i+1  , it suffices to observe that X X is connected follows from the fact that i  is connected and that vertices of □ We stress that if in the proofs of the above three lemmata, we use connected path-decompositions instead of connected tree-decompositions, we obtain the following counterpart of Theorem 1.

| GENERAL PROPERTIES OF OBSTRUCTIONS
We let denote We make the convention that when U u = { }, then we say u-avoiding instead of U -avoiding.
In the rest of this paper, all profs are written using the layout definition of connected treewidth. However, sometimes the reader may find it more intuitive (while less formal) to translate layouts to searching strategies (an equivalence that is formally proved in Section 3). In this sense, it is helpful to see the root vertices of s-pairs and s-triples as ≪departure points≫ of connected search strategies restricted to the corresponding subgraphs.
A vertex v of a graph G is called k-simplicial if it has degree at most k and its neighborhood induces a complete subgraph.
Let v be a cut-vertex of a connected graph G.
and contains a cut-vertex v, then the s-pair G v ( , ) contains exactly two 1-components and v is the unique cut-vertex of G.
Proof. Let v be a cut-vertex of G. Suppose to the contrary that ⩽ , a contradiction to the fact that G k  ∈ . Suppose that G has two distinct cut-vertices, say x and y. Let G x (resp., G y ) be the ) that does not contain y (resp., x) and let C x (resp., C y ) the corresponding components of G x ( , )  (resp., of G y ( , )  ). Notice that, by the discussion above, the choice for G x and G y is unique.
, observe that the graph Ĝ x , respectively, Ĝ y and Ĝ xy , obtained after contracting in G all the edges in G x , respectively, G y and G xy , satisfies x y ctvs ctvs ⩽ ⩽ . As before, from Lemma 6 applied to G G Ĝ ,ˆ,x y y x , we deduce that the following holds: We extract the following cases (see Figure 6):

. Observe that as G x is a contraction of
It is easy to notice that, for every The three cases of the proof of Lemma 7 Lemma 8. For every k 1 ≥ and every connected graph G G , . Suppose to the contrary, that there is a σ H ( ) To that aim, we prove that for any edge e of H 1 , the rooted graph ≤ . Then the conclusion follows from Lemma 1. □ Proof of Claim 1. From Lemma 7, H 1 is biconnected and from Lemma 5 H 1 contains at least three vertices, as otherwise it would contain a pendant vertex which is k-simplicial. This means that J has at least two vertices, therefore v is a cut-vertex also in . It remains to prove that for any proper contraction G′ of G G k , ( ′) ctvs ≤ . To that aim, it suffices to prove the following claim and conclude with Lemma 1.
Claim 2. If G′ is the result of the contraction of some edge e in G, Proof of Claim 2. Assume without loss of generality that e is an edge of H 1 , that is, ≤ , as required.  rooted on the cut-vertex. This structural information will be useful in the later sections. We also observe the following.
has a biconnected underlying graph.
As v is a cut-vertex of H, from Lemma 7, G cannot contain any cut-vertex. □

| OBSTRUCTIONS FOR ctvs AT MOST 2
This section is devoted to the proof of Theorem 3 that characterizes the set 2  . To state this characterization, a few more notations and definitions have to be introduced. First, Figure 7 depicts some rooted graphs that will be central to the description of 2  . Among those, there are three 2-rooted graphs, namely, Observe that R x y and R y x are not isomorphic as their root vertices x and y are switched. Among the 1-rooted graphs described in Figure 7, Y x and Y x (2) are to be distinguished. Indeed, observe that, Figure 8 describes three biconnected graphs. As we will see later, these three graphs are the only biconnected graphs belonging to 2  . From the graphs of Figures 7 and 8, we define the following two sets, respectively, containing 1-rooted graphs and graphs:  be a rooted graph and let S V G ( ) We call the vertices a b , the bases of the 2-twin family S. We say that a graph G G′ = ( ′, ′)  is a 2-twin expansion of G if = ′   and G′ is obtained from G by adding vertices such that each additional vertex is made adjacent with the base vertices of some of the 2-twin families of G. Given a class of rooted graphs  we define its 2-twin expansion ( )  texp as the class of rooted graphs containing all 2-twin expansions of all the graphs in . We say that a rooted graph G is simplified if all its 2-twin families have size 2.
Given a rooted graph G we denote by G the unique simplified rooted graph such that . Given a set  of rooted graphs, we define G G = {˜}   | ∈ . Observe that every graph of 2  is simplified.
All the above definitions apply to graphs as well as we treat graphs as a rooted graph with an empty set of roots.
We are now ready to state the main result of this section.
Outline of the proof of Theorem 3. As a first step we show in Section 5.  ⊕ | ∈ are the non-biconnected graphs in˜2  . The conclusion of the first part appears in Section 5.6. Because of Lemma 8, the non-biconnected case boils down to the identification of˜2 (1)  . This is achieved by considering two cases. If the obstruction contains a separating edge, then we prove that it is the 1-rooted graph Y r , depicted in Figure 7 (this is concluded in Section 5.4). Otherwise, we prove that the obstruction the graph Y x k ( ) , for some k 2 ≥ , that is illustrated in Figure 7 (we prove that in Sections 5.5 and 5.7).

| The set 2
 is closed under 2-twin expansion Lemma 10. Let G be a graph containing a 2-twin family S and such that G ( ) 2 ctvs ≤ . Suppose G contains a path P between the bases a and b of S, such that P is disjoint from S. If G′ is obtained by adding a vertex z adjacent to a and b, then is a 2-twin family of G′. As G ( ) 2 ctvs ≤ , there exists σ such that G σ ( , ) 2 tcost ⩽ . Suppose without loss of generality that σ a σ b ( ) < ( ). Consider σ σ z ′ = ⊙ 〈 〉. Clearly σ′ is a connected layout of G′. Observe that, by construction, for We also observe that, as z is the midvertex of a length-two path between a and thereby that G σ ( , ′) 2 tcost ⩽ . To see this, consider x and y the first two vertices of S in σ (i.e., for be a graph containing a 2-twin family S. If a and b are the bases of S, then G contains a path P from a to b disjoint from S.  . It follows that C a b S { , } ≠ ∪ and thereby G contains a path P from a to b that is disjoint from S. □ Lemma 12. A graph G belongs to 2  iff G belongs to 2  .
Proof. We may assume that G and G are different graphs as otherwise it is trivial. This implies that G contains a 2-twin family of size at least 3 and that for every such family . Let S be a 2-twin family of size at least 3 of G and let a and b be the bases of S. By Lemma 11, G contains a path P from a to b disjoint from S. As G is a 2-twin expansion of G , by Lemma 10, , it remains to show that if e is an edge of G , then G e (˜) 2 ctvs ∕ ≤ . As e is also an edge of G and as G The set X such that G G X = [ ] can be obtained by removing from V G ( ) every vertex z belonging to a 2-twin family T such that T contains two vertices x and y distinct from z with σ x σ y σ z max{ ( ), ( )} < ( ). Observe that e is an edge between two vertices u and v of X . Let w be the vertex resulting from the contraction of e and set X e X u v Then the sequence σ σ X ẽ = ∩ ∕ is a layout of G ẽ∕ . As removing vertices cannot increase the cost of a layout, It remains to prove that the layout σ of G ẽ∕ is connected. This follows from the construction of X . Observe indeed that if, for some vertex u X e S σ u , contains a vertex x X e ∈ ∕ that belongs to the same 2-twin family as z. It follows also contains x and thereby σ is a connected layout of G . Let us now assume that G 2  ∈ . As observed before G contains a 2-twin family S. We prove the statement for V G V G z ( ) = (˜) { } ∪ , that is, there exist two vertices x and y such that S x y z = { , , } is a 2-twin family of G. The fact that G 2  ∈ follows by applying inductively this argument to all the vertices in V G V G ( ) (˜) | ⧹ |. We first prove that G ( ) > 2 ctvs . For the sake of contradiction, suppose that there exists σ G ( ) c  ∈ such that G σ ( , ) 2 tcost ≤ . Observe that one can select such a layout σ such that z is the last vertex of S in σ. Let us consider σ σ σ for some vertex v, then x and y, the twins of z, belong to S σ v ( ( )) σ ( ) t as well and the connectivity of σ − is preserved. It follows that G (˜) 2 ctvs ⩽ : a contradiction to G 2  ∈ . It remains to show that for every edge e E G G e ( ), ( ) 2 ctvs ∈ ∕ ≤ . We now examine two different cases depending on e. We let a and b denote the bases of S.
• Suppose first that e is incident to one of the vertices of S, that is, in G e ∕ vertices a and b are adjacent. Observe that G e ∕ can be obtained from H G e =˜∕ , where ẽ is incident to x or y, by adding the degree-two vertex z adjacent to a and b. As H is a contraction  • Suppose e is not incident to a vertex of S but is not the edge ab. Thereby e is an edge common to G and G and G e ∕ is a twin-expansion of H G e =˜∕ . By Lemma 11, G contains a path from a to b disjoint from S. Observe that H contains such a path P. As G e ∕ is a 2-twin expansion of H and H ( ) 2 ctvs ≤ , by Lemma 10, G e ( ) 2 ctvs ∕ ≤ .
• Suppose e ab = and let v e be the result of the contraction of ab in G . Then G e (˜) 2 ctvs ∕ ≤ and it is easy to see that G e ∕ is the graph obtained from G ẽ∕ after adding in it the vertex z and the edge zv e . Notice that if σ G e (~) Using Lemma 8, we can directly extend Lemma 12 to 1-rooted obstructions: ⇔ ∈ and the lemma follows if we notice that G H = 2 ×˜and apply again Lemma 8. □

| Subsets of (rooted) obstructions
Recall the definition of the sets 2 (1)  and 2  built from the graphs of Figures 7 and 8. We first show that these graphs (and their 2-twin expansions) are 1-rooted obstructions (Lemma 15) and obstructions (Lemma 16), respectively. We will then study the set of 2-rooted obstructions.
). If G is simplified and there exist some graphs (resp., rooted) F H such that F is a contraction of G, then G is isomorphic to H.
Proof. We provide the proof for G H , . For the sake of contradiction, suppose that F is a proper contraction of G.
Proof. From Lemma 13, . Therefore, it is enough to prove that We first prove that G ( ) > 2 ctvs . Observe that for every layout σ G ( ) c  ∈ , there exist adjacent vertices a and b such that a is adjacent to x b , is adjacent to y (i.e., x a b y { , , , } induces a path of G) and σ x σ a σ b σ y ( ) < ( ) < ( ) < ( ). Indeed, this is a direct consequence of the fact that the layout σ is connected and the distance in G between x and y is 3. Observe that G contains three disjoint paths P b from b to y P , a from a to y and P x from x to y. Notice then that S y ( ) contains one vertex from each of P P , a b , and P x , implying that σ G ( , ) 3 tcost ≥ and hence G ( ) > 2 ctvs . Let now consider G x G′ = ( ′, ) 〈 〉 where G′ is the result of the contraction of an edge e in G. We consider two different cases. be the connected components of G x y u − { , , }. Observe that exactly one of these components, say H 0 , contains a single vertex z of degree two in G′ and, for every  of 2-rooted graphs depicted in Figure 9. We say that a biconnected 2-rooted graph . We may assume that H − contains at least two vertices different than x and y. Indeed if this is not the case, we are done as then H x y ( , , ) 〈 〉 is isomorphic either to R xy (if xy E G ( ) ∉ ) or to R xy ). We next prove that H − contains a cut-vertex z. Indeed if this is not the case, then H − contains two internally vertex disjoint paths P 1 and P 2 from x to y.
≤ . These observations permit us to additionally assume that xy E G ( ) ∉ . Notice that every biconnected 2-rooted graph (with at least three vertices) contains G R = xy or G R = xy + as a contraction. Also if G R = xy or G R = xy + the lemma holds trivially. Let G be a 2-rooted graph that is contraction minimal counterexample, that is, Observe that (i)-(iii) hold for G x y G = ( , , ) + + 〈 〉 as well (recall that G + if obtained from G after making its roots adjacent, if they are not already so). We also set Z R G and let P be a path in G between x and y. As G is biconnected (from [i]), all vertices of C are vertices of P. Given a pair a b C C , × 〈 〉 ∈ , we first observe that Z a b ( , , ) is an s-triple and we denote by -avoiding 2-components of Z a b ( , , ). Notice that each G ab is biconnected and is a contraction of G or G + (depending on whether ab E G ( ) ∈ or not). As both G and G + are -free, then each G a b , is also -free. is a proper contraction of G + . As property (iii) holds for G + as well, it holds that, for each i , then one of the following holds: such that z belongs to the subpath of P between a and b.
Proof of Claim 2. We assume that . This implies that G ab is the unique V Z V G ( ) ( ) ⧹ -avoiding 2-component of Z a b ( , , ). As ab E G R ( ), ab + ∉ cannot be isomorphic to G ab . If R xy is isomorphic to G, then the claim follows trivially. We now apply Lemma 17 on Z and G ab and obtain that there is some z C G a b ( ) { , } ab ∈ ⧹ for which (c) holds. □ From Claims 1 and 2 (applied for a x = and b y = ) we obtain that C 3 | | ≥ . Let a x and a y be two vertices of C x y { , } ⧹ with the constraint that a y (resp., a x ) is the one that is closest to the vertex y (resp., x) in P. Notice that a x and a y may be the same vertex.
Proof of Claim 3. Suppose that xa x is an edge of G. The proof for a y y is symmetric. We distinguish two cases (see Figure 10): 〉 be the union of the y-avoiding 2-components of G x a ( , , ) x + . Notice that G x is a proper contraction of G + . Therefore, from (iii), there is a σ G ( ) Let G x ′ be the rooted graph obtained after removing from G all vertices in x ′ ′ 〈 〉 is also a proper contraction of G. Therefore, ≤ , a contradiction to (ii). Case 2. xa x is not a separating edge of G + . Let us consider the graphs G G x = y ⧹ and G G xã = x ∕ . We let v xa x denote the vertex of G resulting from the contraction of the edge xa x . As a x is a cut-vertex and xa x is not a separating edge, there exists an isomorphism from G y to G that maps a V G ( ) Proof. It is easy to verify that for each of the 2-rooted graphs in ( )  texp the value of ctvs is 3, while this value becomes 2 for every proper contraction of a graph of this set. This proves that ( ) Proof. The graphs G and G′ are depicted in Figure 11. We let denote x y a , , , and a′ the vertices of R 2 × xy , and x y z a , , , , and a′ the vertices of R x y (see Figure 11). Observe that G is obtained by the contraction of the edge xz of G′ into the vertex x of G. Let I be the set of vertices distinct from x that belongs to the connected component of H y − containing x. We also define the set of vertices J as the union of the x-avoiding 1-components of H y ( , ) minus the vertex y.
≤ . Without loss of generality, we assume that σ a σ a ( ) < ( ′). We set l σ x σ y = min{ ( ), ( )} and m σ x σ y = max{ ( ), ( )}. Notice that σ a m ( ′) > as otherwise S m ( ) contains a, a′ and a vertex of V H ( ) from a path between x and y. We distinguish four cases. In each of these cases, we build from σ a connected layout σ would contain vertices a, y and a vertex of I , a contradiction. Notice also that the vertices of J do not appear in the tree-supporting set of a vertex in I x a a { , , ′} ∪ . Therefore, we may assume that the vertices y a x a , , , ′ appear consecutively in σ (see Figure 12). We then observe that σ σ y a z x a σ ′ = , , , , ′ l l < + 4 ⊙ 〈 〉 ⊙ ≥ is a connected layout of G′ and that G σ G σ ( ′, ′) = ( , ) 2 tcost tcost ≤ . Case 2. a y x a < < < ′ σ σ σ . Again we can assume that σ a m ( ′) = + 1 and, by the same argument as in the previous case, we deduce that I σ m > +1

⊆
. As σ is connected, and N a x y ( ) = { , }, we have that σ a ( ) = 1. As σ is a connected layout and as y separates J from the rest of the graph, a y < σ implies that y is visited before all the vertices of J . We can thereby assume that J appears after a′ in σ. So we have σ I J = >4 ∪ (see Figure 13). We then observe that σ a y z x a σ ′ = , , , , ′ ∪ . First observe that as x y < σ and y separate the vertices of J from those of I , the connectivity of σ implies that the vertices of J appear after y in σ. So we can assume that J σ m > +1

⊆
. Suppose now that there exists a vertex v I ∈ such that v y < σ . As x is not a cut-vertex, there exists a path P from v to y avoiding x. It then follows that the tree-supporting set S m ( ) . We can assume that σ a σ a m ( ′) = ( ) + 1 = + 2 (see Figure 14). In both cases we observe that σ′ = σ z σ m m > ⊙ 〈 〉 ⊙ ≤ is a connected layout of G′ such that G σ ( ′, ′) = tcost G σ ( , ) 2 tcost ≤ . □ We observe that in the previous lemma, the assumption that x is not a cut-vertex is only used in the third case of the proof. . If H x y ( , , ) 〈 〉 is a 2-component of G x y ( , , ) that is isomorphic to R x y , then x is a cut-vertex of G.  Figure 16, where H is depicted outside the shadow area of the lefthand side graph). We let A a a = { , …, } is an s-triple. But then the edge xa would be a marginal edge (with base z), a contradiction to Lemma 28. □

| Non-biconnected graphs in˜2 
The class of all 2-trees is recursively defined as follows. K 3 is a 2-tree and a graph H with more than three vertices is a 2-tree if it contains some vertex v of degree 2 such that its two neighbors are adjacent and H v ⧹ is a 2-tree. Given a 2-tree G and an edge e E G ( ) ∈ , we say that e is a simplicial edge of G if it is incident to a simplicial vertex.
Lemma 30. Let G be a 2-tree. Then sets of marginal, simplicial, and separating edges of G form a partition of E G ( ).
Proof. Let G be a counterexample with a minimum number of vertices. Clearly G cannot be isomorphic to K 3 as every edge of K 3 is simplicial. A 2-tree that is not isomorphic to K 3 contains a vertex v such that G G v = − ⧹ is a 2-tree. By the minimality of G, the edge set of G − can be partitioned into the sets of the marginal, simplicial, and separating edges. Let x y , be the neighbors of v and let e E G ( ) ∈ . If e vx = or vy, then e is a simplicial edge of G. If e xy = , then it is a separating edge of G. So suppose that e xy vx vy { , , } ∉ . In that case, observe that e has the same type in G as in G − . It follows that the marginal, simplicial, and separating edges of G form a partition of E G ( ). □