Nearly optimal coloring of some $C_{4}$-free graphs

All graphs considered in this paper are finite and simple. We follow [2] for undefined notations and terminologies. Let $G$ be a graph, let $v\in V(G)$, and let $X$ and $Y$ be two subsets of $V(G)$. Let $E(X,Y)$ be the set of edges with one end in $X$ and the other in $Y$. We say that $v$ is complete (resp. anticomplete) to $X$ if $|E(v,X)|=|X|$ (resp. $E(v,X)=\mbox{{\rm\O}}$), and say that $X$ is complete (resp. anticomplete) to $Y$ if each vertex of $X$ is complete (resp. anticomplete) to $Y$.

Let $N_{G}(v)$ be the set of vertices adjacent to $v$, $d_{G}(v)=|N_{G}(v)|$. Let $N_{G}(X)=\{u\in V(G)\setminus X\;|\;u$ has a neighbor in $X\}$, and $N_{G}[X]=X\cup N_{G}(X)$. Let $G[X]$ be the subgraph of $G$ induced by $X$. If it does not cause any confusion, we usually omit the subscript $G$. For $u$, $v\in V(G)$, we write $u\sim v$ if $uv\in E(G)$, and write $u\not\sim v$ if $uv\not\in E(G)$. For $X\subset V(G)$ and $v\in V(G)$, let $N_{X}(v)=N(v)\cap X$.

We say that a graph $G$ contains a graph $H$ if $H$ is isomorphic to an induced subgraph of $G$, and say that $G$ is $H$-free if it does not contain $H$. For a family $\{H_{1},H_{2},\cdots\}$ of graphs, $G$ is $(H_{1},H_{2},\cdots)$-free if $G$ is $H$-free for every $H\in\{H_{1},H_{2},\cdots\}$.

A clique (resp. stable set) of $G$ is a set of mutually adjacent (resp. non-adjacent) vertices in $G$. The clique number (resp. stability number) of $G$, denoted by $\omega(G)$ (resp. $\alpha(G)$), is the maximum size of a clique (resp. stable set) in $G$.

Let $H$ be a graph with $V(H)=\{v_{1},v_{2},\cdots,v_{n}\}$. A clique blowup of $H$ is any graph $G$ such that $V(G)$ can be partitioned into $n$ cliques, say $A_{1},A_{2},\ldots,A_{n}$, such that $A_{i}$ is complete to $A_{j}$ in $G$ if $v_{i}\sim v_{j}$ in $H$, and $A_{i}$ is anticomplete to $A_{j}$ in $G$ if $v_{i}\not\sim v_{j}$ in $H$. Let $G$ be a clique blowup of $H$. We call $G$ a $t$-clique blowup of $H$ if $|A_{i}|=t$ for all $i$, and call $G$ a nonempty clique blowup of $H$ if $A_{i}\neq\mbox{{\rm\O}}$ for all $i$. Under this literature, an induced subgraph of $H$ can be viewed as a clique blowup of $H$ with $A_{i}$ of sizes 1 or 0, for all $i$.

Let $k$ be a positive integer. A $k$-coloring of $G$ is a function $\phi:V(G)\rightarrow\{1,\cdots,k\}$ such that $\phi(u)\neq\phi(v)$ if $u\sim v$. The chromatic number $\chi(G)$ of $G$ is the minimum number $k$ for which $G$ has a $k$-coloring. A graph is perfect if all its induced subgraphs $H$ satisfy $\chi(H)=\omega(H)$. An induced cycle of length at least 4 is called a hole, and its complement is called an antihole. A $k$-hole is a hole of length $k$. A hole or antihole is odd or even if it has odd or even number of vertices. In 2006, Chudnovsky et al [12] proved the Strong Perfect Graph Theorem.

Let ${\cal F}$ be a hereditary family of graphs. If there is a function $f$ such that $\chi(H)\leq f(\omega(H))$ for each graph $H$ in ${\cal F}$, then we say that ${\cal F}$ is $\chi$-bounded, and call $f$ a binding function of ${\cal F}$ [19]. Gyárfás [19], and Sumner [35] independently, conjectured that the class of $T$-free graphs is $\chi$-bounded for any forest $T$. Interested readers are referred to [28, 31, 32] for its related problems and progresses.

Some configurations used in this paper such as paw, diamond, gem, kite, fork, and fork⁺ are shown in Figure 1. Since every antihole with at least six vertices has a 4-hole $C_{4}$, the $\chi$-boundedness of subclasses of $C_{4}$-free graphs are studies extensively. Choudum et al [6] proved $\chi(G)\leq\lceil\frac{5}{4}\omega(G)\rceil$ if $G$ is $(P_{2}\cup P_{3},C_{4})$-free. Chudnovsky et al [10] proved $\chi(G)\leq\frac{3}{2}\omega(G)$ if $G$ is (fork, $C_{4}$)-free. Gasper and Huang [16] proved that $\chi(G)\leq\lfloor\frac{3}{2}\omega(G)\rfloor$ for $(P_{6},C_{4})$-free graph $G$, and Karthick and Maffray [22] improved the upper bound to $\lceil\frac{5}{4}\omega(G)\rceil$. Huang [20] proved that $\chi(G)\leq\lceil\frac{11}{9}\omega(G)\rceil$ if $G$ is $(P_{7},C_{4},C_{5})$-free, which is optimal. Generalize some results of Choudum et al [7], the current authors and Wu proved [4] that $\chi(G)\leq\max\{3,\omega(G)\}$ if $G$ is ($P_{7},C_{4}$, diamond)-free, and $\chi(G)\leq 2\omega(G)-1$ if $G$ is ($P_{7},C_{4}$, gem)-free, and they also proved that $\chi(G)\leq\omega(G)+1$ if $G$ is ($P_{7},C_{4}$, kite)-free.

Motivated by a new concept Polyanna introduced by Chudnovsky et al [8], and by the above mentioned linear binding function of subclasses of $C_{4}$-free graphs, we studied $(C_{4}$, bull)-free graphs and $(C_{4}$, hammer)-free graphs in [5], and proved the following conclusion.

Note that the paths and stars are the two extremal classes of trees. There are a lot of results on binding functions of $P_{t}$-free or $K_{1,3}$-free related graphs. Gyárfás [19] proved that $\chi(G)\leq(t-1)^{\omega(G)-1}$ for all $P_{t}$-free graphs, and this upper bound was improved to $(t-2)^{\omega(G)-1}$ by Gravier et al [18]. On $P_{5}$-free graphs, its best known binding function,due to Scott et al [33], is $\omega(G)^{\log_{2}(\omega(G))}$ for $\omega(G)\geq 3$. Chudnovsky et al [13] proved that a connected $K_{1,3}$-free graphs $G$ with $\alpha(G)\geq 3$ satisfies $\chi(G)\leq 2\omega(G)$. Liu et al [24] proved that $\chi(G)\leq 7\omega^{2}(G)$ if $G$ is fork-free, which answers a problem raised in [21, 28]. Chudnovsky et al [10] proved that $\chi(G)\leq\frac{3}{2}\omega(G)$ if $G$ is (fork, $C_{4}$)-free. Interested readers can find more results and problems on binding functions of subclasses of fork-free graphs in [9, 21, 28, 37].

Let $H$ be a bull or a hammer. From Theorem 1.2, both $(C_{4},P_{7},H)$-free graphs and ($C_{4}$, fork${}^{+},H)$-free graphs are linearly $\chi$-bounded. We can do better on these two classes of graphs. In this paper, we prove the following Theorems 1.3 and 1.4 on the structural decompositions of ($C_{4}$, fork${}^{+},H)$-free graphs and $(C_{4},P_{7},H)$-free graphs, which show that these graphs are closely related to the Petersen graph. Then, we deduce polynomial time algorithms to optimally color these graphs.

Let $F_{0}\sim F_{12}$ be the graphs as shown in Figure 2, where $F_{5}$ is just the Petersen graph, and the parameter $t$ and $t^{\prime}$ appeared in $F_{2}$ and $F^{\prime}_{0}$ are positive integers. When $t=1$, we particularly refer to the graph $F_{2}$ as $F^{\prime}_{2}$. Let $\mathfrak{F}=\{F_{0},F^{\prime}_{0},F_{1},F_{2},\ldots,F_{12}\}$ and, let $\mathfrak{F^{\prime}}=\{F^{\prime}_{2},F_{3},F_{4},F_{5}\}\cup\{C_{k}~{}|~{}k~{}\mbox{is a positive integer of at least 5}\}$. A clique cutset of $G$ is a clique $K$ in $G$ such that $G-K$ has more components than $G$, and a cutvertex is a clique cutset of size 1. A clique $K$ is call a universal clique of $G$ if $K$ is complete to $V(G)\setminus K$.

It is easy to check that $F_{5}-x_{1}$ is isomorphic to $F_{12}-\{x_{4},x_{5}\}$. Since $F_{5}$ is just the Petersen graph, we have that $F_{12}$ can be obtained from the Petersen graph by deleting a vertex and adding a path of length 3 to join some two vertices of degree 3. Since $F_{0}$, $F_{1}$, $F^{\prime}_{2}$, $F_{3}$ and $F_{4}$ are all induced subgraphs of $F_{5}$, and since $F_{6}\sim F_{11}$ are all induced subgraphs of $F_{12}$, Theorems 1.3 and 1.4 assert that ($C_{4}$, fork${}^{+},H)$-free graphs and $(C_{4},P_{7},H)$-free graphs are closely related to the Petersen graph, where $H$ is a bull or a hammer.

Notice that all graphs in $\mathfrak{F^{\prime}}\cup\mathfrak{F^{\prime}}$ are 3-colorable, and clique cutsets and universal cliques are reducible in coloring of graphs. As an immediate consequence of Theorems 1.3 and 1.4, one can prove the following Corollary 1.1 by a simple induction on $|V(G)|$.

Let $k$ be a positive integer. We use $G^{k}$ to denote the $k$-clique blowup of a graph $G$. We can easily verify that both $C_{5}^{k}$ and $C_{7}^{k}$ are ($C_{4},H$, bull)-free for $H$ being a $P_{7}$ or a fork⁺, and $C_{9}^{k}$ is ($C_{4}$, fork⁺, bull)-free. Also, we have that $\chi(C_{5}^{k})\geq\lceil\frac{|V(C_{5}^{k})|}{\alpha(C_{5}^{k})}\rceil=\lceil\frac{5k}{2}\rceil=\lceil\frac{5}{4}\omega(C_{5}^{k})\rceil$, $\chi(C_{7}^{k})\geq\lceil\frac{|V(C_{7}^{k})|}{\alpha(C_{7}^{k})}\rceil=\lceil\frac{7k}{3}\rceil=\lceil\frac{7}{6}\omega(C_{7}^{k})\rceil$, and $\chi(C_{9}^{k})\geq\lceil\frac{|V(C_{9}^{k})|}{\alpha(C_{9}^{k})}\rceil=\lceil\frac{9k}{4}\rceil=\lceil\frac{9}{8}\omega(C_{9}^{k})\rceil$. We will show that by excluding $C_{5}^{k}$ and $C_{7}^{k}$ further, ($C_{4},H$, bull)-free graphs $G$ satisfy $\chi(G)\leq\omega(G)+1$ if $H$ is a $P_{7}$, and $\chi(G)\leq\lceil\frac{9}{8}\omega(G)\rceil$ if $H$ is a fork⁺.

A class ${\cal G}$ of graphs is said to be $\chi$-polydet [31] if ${\cal G}$ has a polynomial binding function $f$ and there is a polynomial time algorithm to determine an $f(\omega(G))$-coloring for each $G\in{\cal G}$.

Suppose that $G$ has a clique cutset $Q$, and suppose that $V(G)\setminus Q$ is partitioned into two nonempty subsets $V_{1}$ and $V_{2}$ that are anticomplete to each other. Let $G_{1}=G[V_{1}\cup Q]$ and $G_{2}=G[V_{2}\cup Q]$. By repeating this procedure to both $G_{1}$ and $G_{2}$ until all the resulted subgraphs have no clique cutsets, $V(G)$ can be partitioned into a collection of subsets of which each induces a subgraph without clique cutsets. We can use a binary tree $T$ to represent this process, and call $T$ a clique-cutsets-decomposition.

Tarjan [34] showed that for a connected graph with $n$ vertices and $m$ edges, its clique-cutsets-decomposition (not necessarily unique) can be found in $O(nm)$ times. Tarjan also showed that if the maximum weight clique of each subgraph induced by a leaf of $T$ can be found, then one can find the maximum weight clique of $G$ in $O(n^{2})$ times.

Chudnovsky et al [11] showed that an optimal coloring of any $C_{4}$-free perfect graph $G$ can be found in $O(|V(G)|^{9})$. If $G$ has $t$ maximal cliques, then we can take $O(t|V(G)|^{3})$ times to find them [26, 36]. Since a $C_{4}$-free graph $G$ has $O(|V(G)|^{2})$ maximal cliques [1, 15], we can find the maximum weight cliques for any $C_{4}$-free graph $G$ in $O(|V(G)|^{5})$ times. Let $F$ be a $P_{7}$ or a fork⁺, and let $H$ be a bull or a hammer. By Theorems 1.3 and 1.4, if $G$ is a $(C_{4},F,H)$-free graph, then in its clique-cutsets-decomposition, each leaf represents a subset of vertices which induces a well-defined graph with a universal clique (maybe empty). Applying Tarjan’s algorithm, we can, in at most $O(|V(G)|^{3})$ times, find a nearly optimal coloring and the maximum weight cliques of a $(C_{4},F,H$)-free graph. Therefore, we have

We will prove Theorem 1.3 in Section 2, prove Theorem 1.4 in Section 3, and prove Theorem 1.5 in Section 4.

This section is devoted to prove Theorem 1.3. We begin from (fork${}^{+},C_{4},C_{3})$-free graphs.

Suppose that $q\geq 3$. Then, $x$ is anticomplete to $\{v_{2},v_{3},v_{2q+1}\}$ to avoid a $C_{3}$ or $C_{4}$, and $x\sim v_{4}$ to avoid an induced fork⁺ on $\{x,v_{1},v_{2},v_{3},v_{4},v_{2q+1}\}$. But then, $x\not\sim v_{5}$, which forces $G[\{x,v_{1},v_{3},v_{4},v_{5},v_{2q+1}\}]$ to be an induced fork⁺, a contradiction. Therefore, $q=2$. During the following proof of Lemma 2.2, every subscript is understood to be modulo 5.

Since $G$ has girth at least 5, we have that each vertex of $N(L)$ has exactly one neighbor in $L$. Let $i\in\{1,\cdots,5\}$. We define $X_{i}=\{x\in N(L)~{}|~{}N_{L}(x)=\{v_{i}\}\}$, and let $X=\bigcup_{i=1}^{5}X_{i}$. Then, $N(L)=X$, and $V(G)=L\cup X\cup R$.

If $E(X_{i},X_{i+1})\neq\mbox{{\rm\O}}$, then $x_{i}v_{i}v_{i+1}x_{i+1}x_{i}$ is a 4-hole for some $x_{i}\in X_{i}$ and $x_{i+1}\in X_{i+1}$. If $X_{i}$ is not complete to $X_{i+2}$, then for some $x^{\prime}_{i}\in X_{i}$ and $x_{i+2}\in X_{i+2}$, $G[\{x^{\prime}_{i},v_{i},v_{i+1},v_{i+2},x_{i+2},v_{i+3}\}]$ is a fork⁺. If $X_{i}$ has two vertices, say $x$ and $x^{\prime}$, then $xv_{1}x^{\prime}x$ is a triangle if $x\sim x^{\prime}$, and $G[\{x,x^{\prime},v_{1},v_{2},v_{3},v_{4}\}]$ is a fork⁺ if $x\not\sim x^{\prime}$. This shows that

In this section, we always assume that $X_{i}=\{x_{i}\}$ if $X_{i}\neq\mbox{{\rm\O}}$. We show next that

Suppose $R\neq\mbox{{\rm\O}}$. We may assume, by symmetry, that $r\sim x_{1}$ for some $r\in R$. Let $Q$ be the component of $G[R]$ that contains $r$. If $X_{3}\neq\mbox{{\rm\O}}$, then $x_{1}\sim x_{3}$ by (1), and so $G[\{r,x_{1},x_{3},v_{1},v_{5},v_{4}\}]$ is a fork⁺. If $X_{4}\neq\mbox{{\rm\O}}$, then $x_{4}\sim x_{1}$ by (1), and so $G[\{r,x_{1},x_{4},v_{1},v_{2},v_{3}\}]$ is a fork⁺. This shows that $X_{3}=X_{4}=\mbox{{\rm\O}}$. Since $x_{1}$ is not a cutvertex of $G$, we may assume by symmetry that $X_{2}=\{x_{2}\}$ and $N_{V(Q)}(x_{2})\neq\mbox{{\rm\O}}$. With a similar argument as above used to deal with $X_{1}$, we can show that $X_{5}=\mbox{{\rm\O}}$. Then, $\{v_{1},v_{2}\}$ is a clique cutset of $G$. Therefore, (2) holds.

Now, we have that $V(G)=L\cup X$, where $X=N(L)\neq\mbox{{\rm\O}}$. Suppose, without loss of generality, that $X_{1}\neq\mbox{{\rm\O}}$. Since $x_{1}$ is anticomplete to $X_{2}\cup X_{5}$ by (1), and since $v_{1}$ is not a cutvertex, we may by symmetry assume that $X_{3}\neq\mbox{{\rm\O}}$. Then, $x_{1}\sim x_{3}$ by (1).

If $X_{4}=X_{5}=\mbox{{\rm\O}}$, then $X_{2}=\mbox{{\rm\O}}$ as otherwise $x_{2}$ is a cutvertex by (1), and thus $G$ is isomorphic to $F_{2}^{\prime}$. Suppose by symmetry that $X_{4}\neq\mbox{{\rm\O}}$. By (1), $x_{1}\sim x_{4}$ and $x_{3}\not\sim x_{4}$. If $X_{2}\neq\mbox{{\rm\O}}$ and $X_{5}\neq\mbox{{\rm\O}}$, then by (1), we have $x_{2}\sim x_{4}$, $x_{2}\sim x_{5}$, and $x_{3}\sim x_{5}$, and thus $G$ is isomorphic to $F_{5}$. If $X_{2}=X_{5}=\mbox{{\rm\O}}$, then $G$ is isomorphic to $F_{3}$. If $X_{2}\neq\mbox{{\rm\O}}$ and $X_{5}=\mbox{{\rm\O}}$, then $x_{2}\sim x_{4}$ by (1), and $G$ is isomorphic to $F_{4}$. The same happens if $X_{2}=\mbox{{\rm\O}}$ and $X_{5}\neq\mbox{{\rm\O}}$. Therefore, $G$ is isomorphic to a graph in $\{F_{3},F_{4},F_{5}\}$. This completes the proof of Lemma 2.2.  

Follows from Lemmas 2.1 and 2.2, we have immediately the following Lemma 2.3.

Suppose that $H$ is a bull, and let $G^{\prime}$ be a (fork${}^{+},C_{4},C_{3})$-free graph such that $G$ is a nonempty clique blowup of $G^{\prime}$. Then, $G^{\prime}$ is a non-complete connected graph without clique cutsets, and so $G^{\prime}$ is in $\mathfrak{F^{\prime}}$ by Lemma 2.3. This completes the proof of Theorem 1.3.  

The aim of this section is to prove Theorem 1.4, which characterizes the structures of $(P_{7},C_{4},bull)$-free graphs and $(P_{7},C_{4},hammer)$-free graphs. First, we discuss $(P_{7},C_{4},C_{3})$-free graphs.

Next, we prove that

Suppose not. Since $G$ is connected, we may assume by symmetry that $a_{1}\in A_{1}$ and $r\in R$ such that $a_{1}\sim r$. Let $Q$ be the component of $G[R]$ which contains $r$. Since $G$ is $P_{7}$-free and $ra_{1}v_{1}v_{2}v_{3}v_{4}$ is an induced $P_{6}$, we have that $a_{1}$ is complete to $V(Q)$. So, $V(Q)=\{r\}$ because $G$ is triangle-free. If $r$ has a neighbor, say $a_{2}$, in $A_{2}$, then $a_{2}ra_{1}v_{1}v_{5}v_{4}v_{3}$ is an induced $P_{7}$ by (3). So, $r$ is anticomplete to $A_{2}$, and is anticomplete to $A_{5}$ by symmetry. If $r$ has a neighbor, say $a_{3}$, in $A_{3}$, then $a_{1}ra_{3}v_{3}v_{4}v_{5}v_{1}a_{1}$ is a 7-hole if $a_{1}\not\sim a_{3}$, and $a_{1}ra_{3}a_{1}$ is a triangle if $a_{1}\sim a_{3}$, both are contradictions. Therefore, $r$ is anticomplete to $A_{3}$, and is anticomplete to $A_{4}$ similarly. Since $G$ is $C_{4}$-free, we have that $a_{1}$ is the unique neighbor of $r$ in $G$, and so is a cutvertex of $G$, a contradiction. This proves (4).

Now, we have that $V(G)=L\cup A$, and $A_{i}\cup A_{i+1}$ is stable by (3). Let $E_{i,i+2}$ be the set of edges between $A_{i}$ and $A_{i+2}$. We show next that for each $i\in\{1,2,\cdots,5\}$,

Let $x\in A_{i}$. It suffices to prove, by symmetry, that $x$ has at most one neighbor in $A_{i+2}$, and if $x$ has a neighbor in $A_{i+2}$ then $A_{i}\setminus\{x\}$ has no neighbor in $A_{i-2}$. If $x$ has two neighbors, say $y$ and $y^{\prime}$, in $A_{i+2}$, then a 4-hole $xyv_{i+2}y^{\prime}x$ appears, a contradiction. If $x$ has a neighbor $y$ in $A_{i+2}$, and some vertex $x^{\prime}\in A_{i}\setminus\{x\}$ has a neighbor $y^{\prime}$ in $A_{i-2}$, then a 7-hole $xyv_{i+2}v_{i-2}y^{\prime}x^{\prime}v_{i}x$ appears. Both are contradictions. Therefore, (5) holds.

Without loss of generality, we take $i=1$, and suppose that $E_{1,3}\neq\mbox{{\rm\O}}$. Choose $a_{1}\sim a_{3}$ with $a_{1}\in A_{1}$ and $a_{3}\in A_{3}$. By (5), $N_{A_{4}}(A_{1}\setminus\{a_{1}\})=\mbox{{\rm\O}}$, and thus $N(A_{1}\setminus\{a_{1}\})\subseteq A_{3}\cup\{v_{1}\}$. Since $A_{1}$ is a stable set and $v_{1}$ is not a cutvertex, we have that each vertex in $A_{1}\setminus\{a_{1}\}$ must have a neighbor in $A_{3}$. Similarly, each vertex in $A_{3}\setminus\{a_{3}\}$ must have a neighbor in $A_{1}$. Therefore, $|A_{1}|=|A_{3}|$, and so $E_{1,3}$ is a matching of size $|A_{1}|$ by (5). This proves (6).

Recall that $V(G)=L\cup A$. If $A=\mbox{{\rm\O}}$ then $G$ is isomorphic to $F_{1}$. So, we suppose that $A\neq\mbox{{\rm\O}}$, and suppose, without loss of generality, that $A_{1}\neq\mbox{{\rm\O}}$. By (5), $N(A_{1})\subseteq A_{3}\cup A_{4}\cup\{v_{1}\}$. Since $v_{1}$ is not a cutvertex, we may assume by symmetry that $N(A_{1})\cap A_{3}\neq\mbox{{\rm\O}}$. Then, by (5) and (6), $|A_{1}|=|A_{3}|=|E_{1,3}|\geq 1$ and $E_{1,3}$ is a matching.

Let $a_{1}\in A_{1}$ and $a_{3}\in A_{3}$ such that $a_{1}\sim a_{3}$. We first consider the case that $E_{4,1}\neq\mbox{{\rm\O}}$.

First, we suppose that $E_{3,5}\neq\mbox{{\rm\O}}$. By (5) and (6), $|A_{5}|=|A_{3}|=|E_{3,5}|=1$. Let $A_{5}=\{a_{5}\}$. Now, we have that $V(G)=\{a_{1},a_{3},a_{4},a_{5}\}\cup A_{2}\cup L$ such that $G[\{a_{1},a_{3},a_{4},a_{5}\}]=a_{5}a_{3}a_{1}a_{4}$ by (3). If $A_{2}=\mbox{{\rm\O}}$, then $G$ is isomorphic to $F_{4}$. Otherwise, let $a_{2}\in A_{2}$. By (3), $N(a_{2})\subseteq\{a_{4},a_{5},v_{2}\}$, and so $a_{2}$ must have a neighbor in $\{a_{4},a_{5}\}$ since $v_{2}$ cannot be a cutvertex. Without loss of generality, we suppose $a_{2}\sim a_{4}$. Then, $a_{2}\sim a_{5}$ for avoiding an induced $P_{7}=v_{2}a_{2}a_{4}a_{1}a_{3}a_{5}v_{5}$, and thus $|A_{2}|=|A_{5}|=|E_{5,2}|=1$ by (6), which forces that $G$ is isomorphic to $F_{5}$.

Similarly, we may deduce that $G$ is isomorphic to $F_{4}$ or $F_{5}$ if $E_{2,4}\neq\mbox{{\rm\O}}$. Next, we suppose that $E_{2,4}=E_{3,5}=\mbox{{\rm\O}}$.

If $A_{2}\neq\mbox{{\rm\O}}$, let $a_{2}\in A_{2}$, then $N(a_{2})\subseteq\{v_{2}\}\cup A_{5}$ by (3), and so $a_{2}$ must have a neighbor, say $a_{5}$, in $A_{5}$ as otherwise $v_{2}$ is a cutvertex. But now, $v_{2}a_{2}a_{5}v_{5}v_{4}a_{4}a_{1}$ is an induced $P_{7}$ as $a_{5}\not\sim a_{1}$ and $a_{5}\not\sim a_{4}$ by (3). A similar contradiction happens if $A_{5}\neq\mbox{{\rm\O}}$. Therefore, $A_{2}=A_{5}=\mbox{{\rm\O}}$, and $V(G)=\{a_{1},a_{3},a_{4}\}\cup L$. Now we have that $G$ is isomorphic to $F_{3}$. This proves Claim 3.1.  

By Claim 3.1 and by symmetry, we may suppose that $E_{4,1}=E_{3,5}=\mbox{{\rm\O}}$. If $A_{4}\neq\mbox{{\rm\O}}$, let $a_{4}\in A_{4}$, then by (3), and by the fact that $v_{4}$ cannot be a cutvertex, $a_{4}$ must have a neighbor, say $a_{2}$, in $A_{2}$ which forces an induced $P_{7}=v_{4}a_{4}a_{2}v_{2}v_{1}a_{1}a_{3}$, a contradiction. So, $A_{4}=\mbox{{\rm\O}}$. Similarly, $A_{5}=\mbox{{\rm\O}}$, and thus $A_{2}=\mbox{{\rm\O}}$ as otherwise $v_{2}$ is a cutvertex. Now, we have that $V(G)=A_{1}\cup A_{3}\cup L$, $|A_{1}|=|A_{3}|\geq 1$, and $E_{1,3}$ is a matching of size $|A_{1}|$. Therefore, $G$ is isomorphic to $F_{2}$. This completes the proof of Lemma 3.1.