Minimum Partition of Polygons Under Width and Cut Constraints

Damian and Pemmaraju [14] and Damian-Iordache [15] gave a polynomial-time algorithm for partitioning a simple polygon into the minimum number of subpolygons without using Steiner points such that each subpolygon has diameter at most $\alpha$, for $\alpha >0$. Later, Buchin and Selbach [11] showed that this problem becomes NP-hard for polygons with holes. Worman [22] proved NP-completeness for a variant in which each subpolygon must be contained in an axis-aligned square of side length $\alpha$. Abrahamsen and Stade [2] showed that allowing Steiner points leads to NP-hardness for the partition problem under axis-aligned unit-square containment, even for simple polygons without holes. This marks the first known NP-hardness result of the minimum partition problems for hole-free polygons. Abrahamsen and Rasmussen [1] studied the problem of partitioning simple polygons into the minimum number of pieces such that each piece satisfies a bounded-size constraint (e.g., unit area, perimeter, diameter, or containment within unit disks or squares).

Our main contribution is an analysis of the minimum partition number under constraints $W,U\subseteq {𝕊}^{+}$. First, we provide necessary and sufficient conditions for the existence of feasible partitions in $\text{wPartition}(P,W,U)$, along with a decision algorithm for testing feasibility

Second, we study the monotonicity of the minimum partition number under polygon containment $Q\subseteq P$ (Sections 3, 4, and 5). We show that this monotonicity does not hold in general, and identify a sufficient condition based on a restricted-orientation convexity, called $𝒪$-convexity, where $𝒪\subseteq {𝕊}^{+}$. Theorem 3 states that $\text{opt}(Q,W,U)\le \text{opt}(P,W,U)$ holds if $Q$ is $U$-convex with respect to $P$ for guillotine partitions, or $\overline{W}$-convex with respect to $P$ for non-guillotine partitions, where $\overline{W}$ is the set of all unit vectors perpendicular to those in $W$.

Finally, we prove a partition analogue of Bang’s conjecture (Section 6). The statement of Bang’s conjecture is as follows: if a convex body $K\subset {ℝ}^d$ is covered by strips $H_1,\mathrm{\dots },H_m$, then ${\sum }_{i=1}^m{inf}_{𝐯\in {𝕊}^{+}}\frac{{\omega }_{𝐯}(H_i)}{{\omega }_{𝐯}(K)}\ge 1$. Our theorem replaces strip coverings with arbitrary partitions and extends the direction set to any subset $W\subseteq {𝕊}^{+}$.

To the best of our knowledge, this is the first partition analogue that allows non-convex pieces. We also show that, for $U\subseteq \overline{W}$, an optimal partition of a convex polygon can be computed in linear time using equally spaced parallel cuts (See Corollary 17). The omitted proofs will be found in the full version of the paper at https://arxiv.org/abs/2509.09981.

For $C_j\in R_i$, $C_j$ is a subpolygon of $Q$ that touches $\partial Q$. The intersection $\partial C_j\cap \partial Q$ consists of continuous paths on $\partial Q$. Specifically, it can be the loop $\partial Q$ itself if $\partial Q\subseteq \partial C_j$, implying that $Q$ satisfies unit-width constraint $W$. However, assuming $\text{opt}(Q)>1$, this case does not occur. Thus, every connected component of $\partial C_j\cap \partial Q$ must be a continuous path on $\partial Q$ that is not a loop. We define ${ℐ}_{ij}$ as

We then define the aggregate sets ${ℐ}_i≔{\bigcup }_{j=1}^t{ℐ}_{ij}$ and ${ℐ}_Q≔{\bigcup }_{i=1}^m{ℐ}_i$.

Let $I\in {ℐ}_{ij}$ be a continuous path along $\partial Q$ with endpoints $p$ and $q$ such that $I$ corresponds to the portion of $\partial Q$ from $p$ to $q$ in counterclockwise order. Since $\partial Q$ is a simple closed curve, topologically equivalent to a circle, we regard each path $I$ as a circular interval on $\partial Q$. As ${ℐ}_{ij}$ contains no loops, the case $p=q$ occurs only when $\partial C_j$ touches $\partial Q$ at a single point $p$, and no other portion of $\partial C_j$ near $p$ intersects $\partial Q$. Such intervals are called degenerate intervals.

For $i\in [m]$, ${ℐ}_i$ may contain both degenerate and non-degenerate intervals. Since $P_i$ is simple, all intervals in ${ℐ}_i$ are pairwise interior-disjoint. Figure 5(a) shows a piece $P_i$ and the subpolygon $Q$, from which the circular intervals in ${ℐ}_i$ are defined. These intervals are illustrated in Figure 5(b).

Note that a degenerate interval at some point $p\in \partial Q$ may appear in multiple ${ℐ}_i$’s. In such cases, each occurrence is treated as a distinct element in ${ℐ}_Q$. Since $\mathrm{\Pi }=\{P_1,\mathrm{\dots },P_m\}$ is a partition of $P$ and $Q\subseteq P$, every point on $\partial Q$ belongs to some interval of ${ℐ}_Q$, and no two intervals in ${ℐ}_Q$ intersect each other in their interiors. Thus, ${ℐ}_Q$ forms a decomposition of $\partial Q$.

Finally, we relate these circular intervals to layer segments so as to define a link instruction formally. For any $j\in [t]$ and any $k\in [h]$, consider the component $C_j\in R_i$ and the layer $L_k$. Each connected component of $𝒜\cap C_j$ (or $L_k\cap C_j$) is bounded by four parts: continuous portions of the inner and outer boundaries of $𝒜$ (or $L_k$), and two edges from $E_{ij}^{\mathrm{bd}}$. Each connected component of $L_k\cap C_j$ is entirely contained within a unique connected component of $𝒜\cap C_j$. Moreover, the portion of $\partial Q$ that bounds each connected component of $𝒜\cap C_j$ corresponds to a circular interval in ${ℐ}_{ij}$. Thus, each layer segment of $C_j$ in $L_k$ is uniquely associated with a circular interval in ${ℐ}_{ij}$, in a one-to-one correspondence. The number of layer segments in $𝒜\cap C_j$ (or $L_k\cap C_j$) is $|{ℐ}_{ij}|\cdot h$ (or $|{ℐ}_{ij}|$).

We revisit the link instruction that merges two layer segments $Z_1\subseteq L_k\cap C_j$ and $Z_2\subseteq L_k\cap C_{j^{\prime }}$ for $j,j^{\prime }\in [t]$ along layer $L_k$. The link instruction reallocates the region in $L_k$ spanned counterclockwise from $Z_1$ to $Z_2$. Since $Z_1$ and $Z_2$ correspond to circular intervals in ${ℐ}_{ij}$ and ${ℐ}_{i{j}^{\prime }}$, respectively, each link instruction can be represented by a triple $(I_1,I_2,k)$, where $I_1,I_2\in {ℐ}_Q$ and $k\in [h]$ (See Figure 4). Note that $(I_1,I_2,k)\ne (I_2,I_1,k)$, as the counterclockwise span from $Z_1$ to $Z_2$ differs from that in the reverse order.

For each $C_j\in R_i$, $\partial C_j\cap \text{int}(Q)$ consists of connected components, each forming a continuous path connecting two points on $\partial Q$. By the definition of ${ℐ}_{ij}$, the endpoints of these paths correspond to the endpoints of the circular intervals in ${ℐ}_{ij}$. The total number of such paths in $\partial C_j\cap \text{int}(Q)$ is $|{ℐ}_{ij}|$. See Figure 5(a).

We traverse $\partial P_i$ counterclockwise, starting at any point on $\partial C_1\cap \text{int}(Q)$ and completing a full circuit. During the traversal, each path in $\partial C_j\cap \text{int}(Q)$ is visited exactly once for each $j\in [t]$, except for the path containing the starting point. The order in which the continuous paths are visited is denoted by ${\gamma }_1\to {\gamma }_2\to \mathrm{\cdots }\to {\gamma }_l$, where ${\gamma }_1={\gamma }_l$ is the path in $\partial C_1\cap \text{int}(Q)$ containing the starting point, and $l={\sum }_{j=1}^t|{ℐ}_{ij}|+1=|{ℐ}_i|+1$.

Consider two consecutive paths ${\gamma }_k$ and ${\gamma }_{k+1}$ for any $k\in [l-1]$. Each path is derived from some piece in $R_i$; there exist $j,j^{\prime }\in [t]$ such that ${\gamma }_k\subseteq \partial C_j\cap \text{int}(Q)$ and ${\gamma }_{k+1}\subseteq \partial C_{j^{\prime }}\cap \text{int}(Q)$. As we traverse from ${\gamma }_k$ to ${\gamma }_{k+1}$, we exit $\text{int}(Q)$ through an endpoint of ${\gamma }_k$ and re-enter $\text{int}(Q)$ through an endpoint of ${\gamma }_{k+1}$.

By Lemma 5, a pair of distinct intervals in ${ℐ}_i^{+}$ provides an exit-entry pair for each transition between distinct pieces in $R_i$ in the counterclockwise traversal of $\partial P_i$.

Based on the counterclockwise traversal of $\partial P_i$, we construct a directed graph $G_i=(V_i,E_i)$ as follows. Each vertex $v\in V_i$ corresponds to an interval in ${ℐ}_i^{+}$, so $|V_i|=|{ℐ}_i^{+}|$. Let ${\gamma }_1$ and ${\gamma }_2$ be consecutive paths in the traversal where ${\gamma }_1\subseteq \partial C_j\cap \text{int}(Q)$ and ${\gamma }_2\subseteq \partial C_{j^{\prime }}\cap \text{int}(Q)$ with $j\ne j^{\prime }$. The traversal from ${\gamma }_1$ to ${\gamma }_2$ encounters intervals $I_1$ and $I_2$ in ${ℐ}_i^{+}$ that contain the exit and entry points, respectively. By Lemma 5, these intervals $I_1$ and $I_2$ are uniquely determined. We add a directed edge $e$ between vertices $v_1$ and $v_2$ corresponding to $I_1$ and $I_2$, respectively.

Let $p$ and $q$ denote the exit and entry points of the traversal, respectively; $p$ is the endpoint of $I_1$ and $q$ is the endpoint of $I_2$. The counterclockwise traversal on $\partial P_i$ between ${\gamma }_1$ and ${\gamma }_2$ follows a simple path along $\partial P_i$ outside $Q$ that starts from $p$ and ends at $q$. Let ${\gamma }_{pq}\subseteq \partial P_i\setminus \text{int}(Q)$ denote this path from $p$ to $q$. Observe that the exit and entry points may coincide, i.e., $p=q$, if $I_1$ and $I_2$ are adjacent at $p$ and $I_1$ lies counterclockwise from $I_2$ along $\partial Q$. In this case, the direction of $e$ is assigned from $v_2$ to $v_1$.

Given that $p\ne q$, the simple path ${\gamma }_{pq}$ can be viewed as a simple path connecting two distinct points on the boundary of the circle and lying outside the circle. The path ${\gamma }_{pq}$ can be classified into one of two types, depending on how it winds around the circle. Formally, ${\gamma }_{pq}$ is homotopic to a directed path in ${ℝ}^2\setminus \text{int}(Q)$ that winds around $\partial Q$ in either counterclockwise or clockwise direction. Note that it cannot wind around the boundary more than once since ${\gamma }_{pq}$ is simple. We assign the direction of $e$ from $v_1$ to $v_2$ if the path is of the counterclockwise type, and from $v_2$ to $v_1$, otherwise. Figure 5(b) illustrates ${\gamma }_{pq}$ for both cases where $p=q$ and $p\ne q$. The same rule is applied to assign directions to all other edges in $G_i$.

In summary, we construct the directed graph $G_i$ for $i\in [m]$ to represent how the disjoint components of $R_i$ are to be connected into a single piece within $Q$. Each vertex corresponds to an interval in ${ℐ}_i^{+}$. Each directed edge $e=(v_1,v_2)$ of $G_i$ represents a link instruction $(I_1,I_2,k)$, but $k$ is not specified yet. See Figure 5(c) for an illustration of the directed graph $G_i$ that defines three link instructions, where $P_i$ and $Q$ are as shown in Figure 5(a).

The graph $G_{𝒥}$ represents the transitive closure of the proper containment relation. That is, for $J_1,J_2\in {𝒥}_Q$, there is a directed edge from $J_1$ to $J_2$ in $G_{𝒥}$ if there exists some $J^{\prime }\in {𝒥}_Q$ such that $(J_1,J^{\prime })$ and $(J^{\prime },J_2)$ are directed edges in $G_{𝒥}$. A transitive reduction of $G_{𝒥}$ is a directed graph on the same vertex set with the minimum number of edges that preserves all reachability relations of $G_{𝒥}$. Since the transitive reduction of a DAG is unique [8], we denote the transitive reduction of $G_{𝒥}$ by $G_{𝒥}^{\mathrm{tr}}$. Let $U{G}_{𝒥}^{\mathrm{tr}}$ denote the underlying graph of $G_{𝒥}^{\mathrm{tr}}$.

By Lemma 9, $U{G}_{𝒥}^{\mathrm{tr}}$ is a forest. Each directed tree $T$ of $G_{𝒥}^{\mathrm{tr}}$ has total out-degrees $|T|-1$, implying that exactly one vertex has out-degree zero and all others have out-degree one. This structure corresponds to an in-branching tree [8].

In each in-branching tree of $G_{𝒥}^{\mathrm{tr}}$, the unique vertex with out-degree zero is called the root. The level of a vertex is defined as the number of edges on the path from the root to that vertex plus one, where the level of the root is one. The level of each interval $J\in {𝒥}_Q$ is defined to be the level of the corresponding vertex in $G_{𝒥}^{\mathrm{tr}}$.

Recall that each circular interval $J\in {𝒥}_Q$ is formed by merging intervals in ${ℐ}_i^{+}$ for some $i\in [m]$. These intervals correspond to the vertices of a connected component $T$ of $G_i$, where each edge of $T$ encodes a link instruction that merges two elements in $R_i$. Let $\mathrm{𝙸𝙽𝚂𝚃}(J)$ denote the set of link instructions corresponding to the edges of $T$. We assign the $k$-th layer to every link instruction in $\mathrm{𝙸𝙽𝚂𝚃}(J)$ if $J$ is at level $k$ in $G_{𝒥}^{\mathrm{tr}}$. Note that the maximum level in $G_{𝒥}^{\mathrm{tr}}$ is at most $\lfloor |{ℐ}_Q^{+}|/2\rfloor$ if each connected component of $G_i$ has size two for all $i=1,2,\mathrm{\dots },m$. Thus, we set the number of layers in the corridor $𝒜$ to this maximum level: $h=\lfloor |{ℐ}_Q^{+}|/2\rfloor$.

For each $J\in {𝒥}_Q$, all link instructions in $\mathrm{𝙸𝙽𝚂𝚃}(J)$ reallocate layer segments within a single layer $L_k$ to a common piece $P_i$, where $i\in [m]$ and $k\in [h]$ are determined by $J$. Let $I_a,I_b\in {ℐ}_i^{+}$ such that $J$ is the smallest circular interval that spans from $I_a$ to $I_b$ in counterclockwise order. Let $Z_a,Z_b\subseteq L_k$ be the layer segments corresponding to $I_a$ and $I_b$, respectively. Applying $\mathrm{𝙸𝙽𝚂𝚃}(J)$ is equivalent to reallocating every $Z\in L_k[Z_a,Z_b]$ to $P_i$.

In summary, ${\mathrm{𝙸𝙽𝚂𝚃}}_Q={\bigcup }_{J\in {𝒥}_Q}\mathrm{𝙸𝙽𝚂𝚃}(J)$ defines the set of all link instructions for reconfiguration. Applying ${\mathrm{𝙸𝙽𝚂𝚃}}_Q$ to $\mathrm{\Pi }[Q]$ yields a partition $Q=Q_1^{\ast }\cup \mathrm{\cdots }\cup Q_m^{\ast }$, where each $Q_i^{\ast }$ is a region assigned to $P_i$ within $Q$. When $Q$ is $\overline{W}$-convex with respect to $P$, each $Q_i^{\ast }$ is connected and satisfies both constraints $W$ and $U$.

We show that any two pieces $C_j,C_{j^{\prime }}\in R_i$ are connected by some link instructions in ${\mathrm{𝙸𝙽𝚂𝚃}}_i$. During the traversal on $\partial P_i$, each continuous path of $\partial C_j\cap \text{int}(Q)$ is encountered at least once for every $j\in [t]$, and we denote the sequence of these paths in the order they are visited as $({\gamma }_1,{\gamma }_2,\mathrm{\dots },{\gamma }_l)$, where ${\gamma }_1={\gamma }_l$. Then, there exist indices $k,k^{\prime }\in [l]$ such that ${\gamma }_k\subseteq \partial C_j\cap \text{int}(Q)$ and ${\gamma }_{k^{\prime }}\subseteq \partial C_{j^{\prime }}\cap \text{int}(Q)$. Since it is a cyclic sequence with ${\gamma }_1={\gamma }_l$, we assume without loss of generality that .

Recall that an edge of $G_i$ is added whenever two consecutive paths in the sequence are derived from distinct elements in $R_i$. This indicates that a transition between them occurs during the traversal. The link instruction associated with this edge merges the corresponding elements in $R_i$. The subsequence $({\gamma }_k,{\gamma }_{k+1},\mathrm{\dots },{\gamma }_{k^{\prime }})$ contains multiple transitions between distinct elements in $R_i$, starting from $C_j$ and eventually reaching $C_{j^{\prime }}$. Applying all link instructions associated with the transitions in $({\gamma }_k,\mathrm{\dots },{\gamma }_{k^{\prime }})$ results in $C_j$ and $C_{j^{\prime }}$ being merged into a single connected piece.

Let $Q_i$ denote the subregion of $Q$ resulting from merging elements of $R_i$ via ${\mathrm{𝙸𝙽𝚂𝚃}}_i$, with no other instructions applied. Then, $Q_i$ consists of all elements in $R_i$ and layer segments reallocated from other pieces: $Q_i=\left({\bigcup }_{j\in [t]}{C}_j\right)\cup {𝒵}_i^{\mathrm{out}}$, where ${𝒵}_i^{\mathrm{out}}$ is the set of layer segments reallocated to $P_i$ by ${\mathrm{𝙸𝙽𝚂𝚃}}_i$.

The region $𝐋=L_1\cup \mathrm{\cdots }\cup L_h$ denotes the union of all layers in $Q$. For each $C_j\in R_i$, $𝐋\cap C_j$ consists of layer segments within $C_j$. By construction of layers, $C_j\setminus 𝐋$ is connected and non-empty. We refer to this region as a core of $C_j$, denoted by $\mathrm{core}(C_j)=C_j\setminus 𝐋$. Each $C_j$ consists of its core together with the layer segments it contains. Let ${𝒵}_i^{\mathrm{in}}$ denote the set of all layer segments in $𝐋\cap C_j$ over all $C_j\in R_i$. Therefore, for $i\in [m]$, we have $Q_i=\left({\bigcup }_{j\in [t]}\mathrm{core}(C_j)\right)\cup {𝒵}_i^{\mathrm{in}}\cup {𝒵}_i^{\mathrm{out}}.$ Figure 7(a) illustrates the parts of $Q_i$: cores and layer segments. The shapes are drawn schematically to reflect the topological structure, rather than an exact polygonal description.

We now turn to the link instructions in ${\mathrm{𝙸𝙽𝚂𝚃}}_Q\setminus {\mathrm{𝙸𝙽𝚂𝚃}}_i$ that are applied to $Q_i$. Let $Q_i^{\ast }$ be a subregion of $Q_i$ that is obtained by applying all link instructions in ${\mathrm{𝙸𝙽𝚂𝚃}}_Q\setminus {\mathrm{𝙸𝙽𝚂𝚃}}_i$ to $Q_i$. Note that any part of $Q_i$ reallocated by ${\mathrm{𝙸𝙽𝚂𝚃}}_Q\setminus {\mathrm{𝙸𝙽𝚂𝚃}}_i$ lies within ${𝒵}_i^{\mathrm{in}}\cup {𝒵}_i^{\mathrm{out}}$, and thus the cores remain unchanged. We show that $Q_i^{\ast }$ is well-defined, meaning that $Q_i^{\ast }$ is invariant under the order in which the instructions in ${\mathrm{𝙸𝙽𝚂𝚃}}_Q$ are applied.

Lemma 10 implies that no layer segment is reassigned by more than one link instruction. As a consequence, we obtain the following corollary, which states that $Q_i^{\ast }$ is well-defined.

Corollary 11 ensures that each layer segment may be reallocated to a different piece at most once, and no chains of reallocations such as $P_{i_1}\to P_{i_2}\to P_{i_3}$ with $i_1\ne i_2$ and $i_2\ne i_3$ occur.

This lemma further implies that layer segments in ${𝒵}_i^{\mathrm{out}}$ are preserved, while only those in ${𝒵}_i^{\mathrm{in}}$ are reallocated by ${\mathrm{𝙸𝙽𝚂𝚃}}_Q\setminus {\mathrm{𝙸𝙽𝚂𝚃}}_i$. Let ${𝒵}_i^{\mathrm{in}\ast }$ be the subset of ${𝒵}_i^{\mathrm{in}}$ that consists of layer segments preserved under ${\mathrm{𝙸𝙽𝚂𝚃}}_Q\setminus {\mathrm{𝙸𝙽𝚂𝚃}}_i$. Then, $Q_i^{\ast }$ is the subpolygon that is obtained from $Q_i$ by removing those layer segments in ${𝒵}_i^{\mathrm{in}}\setminus {𝒵}_i^{\mathrm{in}\ast }$. Then $Q_i^{\ast }=\left({\bigcup }_{j\in [t]}\mathrm{core}(C_j)\right)\cup {𝒵}_i^{\mathrm{in}\ast }\cup {𝒵}_i^{\mathrm{out}},$ where ${𝒵}_i^{\mathrm{in}\ast }\subseteq {𝒵}_i^{\mathrm{in}}$. Figure 7(b–c) illustrates the construction of $Q_i^{\ast }$, in which only layer segments in ${𝒵}_i^{\mathrm{in}}$ are reallocated by ${\mathrm{𝙸𝙽𝚂𝚃}}_Q\setminus {\mathrm{𝙸𝙽𝚂𝚃}}_i$.

To prove that $Q_i^{\ast }$ forms a connected piece, it suffices to verify two types of path-connectivity among its constituent parts, which must be preserved during the reallocation induced by ${\mathrm{𝙸𝙽𝚂𝚃}}_Q\setminus {\mathrm{𝙸𝙽𝚂𝚃}}_i$.

• $\mathrm{■}$

  Each layer segment in ${𝒵}_i^{\mathrm{in}\ast }\cup {𝒵}_i^{\mathrm{out}}$ is path-connected to some $\mathrm{core}(C_j)$ within $Q_i^{\ast }$ for $C_j\in R_i$.

• $\mathrm{■}$

  The cores $\{\mathrm{core}(C_j)\mid C_j\in R_i\}$ are mutually path-connected within $Q_i^{\ast }$.

Here, two sets $A,B\subseteq X$ are said to be path-connected within a region $X$ if there exists a path in $X$ joining some $a\in A$ and $b\in B$.

By Lemma 12, every layer segment in ${𝒵}_i^{\mathrm{in}\ast }\cup {𝒵}_i^{\mathrm{out}}$ has a path to some core within $Q_i^{\ast }$. It remains to show that $\mathrm{core}(C_1),\mathrm{\dots },\mathrm{core}(C_t)$ are mutually path-connected within $Q_i^{\ast }$.

By Lemmas 12 and 13, each $Q_i^{\ast }$ is connected. Thus, applying ${\mathrm{𝙸𝙽𝚂𝚃}}_Q$ to $\mathrm{\Pi }[Q]$ yields the partition ${\mathrm{\Pi }}^{\ast }[Q]=\{Q_1^{\ast },Q_2^{\ast },\mathrm{\dots },Q_m^{\ast }\}$, where each $Q_i^{\ast }$ is a connected subregion of $Q$.

Applying link instructions in ${\mathrm{𝙸𝙽𝚂𝚃}}_Q$ may induce holes within merged pieces in the reconfigured partition of $Q$. When $Q_i^{\ast }$ contains holes, reallocating them to $P_i$ does not increase ${\omega }_{𝐯}(Q_i^{\ast })$ for every $𝐯\in {𝕊}^{+}$. Therefore, each $Q_i^{\ast }$ can be regarded as a simple polygon.

The region ${𝐙}_{\lambda }$ is connected and bounded by four parts: two continuous portions of the inner and outer boundaries of $𝒜$, and two subsegments of edges from $C_j$ and $C_{j^{\prime }}$. The outer boundary of $𝒜$ is $\partial Q$ and its inner boundary is $\partial Q^{\varphi }$, where $Q^{\varphi }$ is the inner $\varphi$-offset polygon of $Q$.

For the sake of clarity, we introduce interval notation to represent subpaths of $\partial Q$ and $\partial Q^{\varphi }$. For any two points $x,y\in \partial Q$, we denote by $\partial Q[x,y]$ the portion of $\partial Q$ from $x$ to $y$ in counterclockwise order, including both endpoints. Let $\partial Q(x,y]=\partial Q[x,y]\setminus \{x\}$, $\partial Q[x,y)=\partial Q[x,y]\setminus \{y\}$, and $\partial Q(x,y)=\partial Q[x,y]\setminus \{x,y\}$. Similarly, we define $\partial Q^{\varphi }[x,y],\partial Q^{\varphi }(x,y],\partial Q^{\varphi }[x,y)$, and $\partial Q^{\varphi }(x,y)$ as portions of $\partial Q^{\varphi }$.