# Structures of Critical Nontree Graphs with Cutwidth Four

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Lemma**

**1.**

- (1)
- If H is a subgraph of G, then $c\left(H\right)\le c\left(G\right)$.
- (2)
- If H is homeomorphic to G, then $c\left(H\right)=c\left(G\right)$.
- (3)
- For a cut edge e in G, if ${V}_{1},{V}_{2}$ are the vertex sets of two components of $G-e$, then there exists an optimal labeling ${f}^{*}$, such that the vertices in each of ${V}_{1}$ and ${V}_{2}$ are labeled consecutively.

**Lemma**

**2**

**Lemma**

**3**

**Lemma**

**4**

## 2. Preliminary Results

**Definition**

**1.**

- (i)
- For graph G and integer $r>0$, let $v\in V\left(G\right)$ with ${d}_{G}\left(v\right)>r$. For ${v}_{1},{v}_{2},\dots ,{v}_{r}\in {N}_{G}\left(v\right)$, define $G(v;{v}_{1},{v}_{2},\dots ,{v}_{r})$ to be the component of $G-\{v{v}_{1},v{v}_{2},\dots ,v{v}_{r}\}$ that contains v.
- (ii)
- Let ${G}_{1},{G}_{2}$ be two disjoint graphs with $u\in V\left({G}_{1}\right)$ and $v\in V\left({G}_{2}\right)$. To identify u and v, denoted as ${G}_{1}{\oplus}_{u,v}{G}_{2}$, is to replace $u,v$ by a single vertex z$(i.e.,u=v=z)$ incident to all the edges which were incident to u and v, where z is called the identified vertex.
- (iii)
- Let ${G}_{1}$, ${G}_{2}$ and ${G}_{3}$ be three disjoint graphs, ${D}_{3}\left({K}_{1,3}\right)=\left\{{u}_{0}\right\}$ and ${D}_{1}\left({K}_{1,3}\right)=\{{u}_{1},{u}_{2},{u}_{3}\}$, ${v}_{j}\in V\left({G}_{j}\right)$ for each $j\in \mathcal{S}$${}_{3}$. Define ${K}_{1,3}\circ ({G}_{1},{G}_{2},{G}_{3})$ as the graph obtained from the disjoint union ${G}_{1},{G}_{2},{G}_{3}$ and ${K}_{1,3}$ by identifying ${u}_{j}$ with ${v}_{j}$ (again denoted as ${v}_{j}$) for each $j\in \mathcal{S}$${}_{3}$ (see Figure 3d in Section 3.1 below).
- (iv)
- Let ${G}_{1}$, ${G}_{2}$ and ${G}_{3}$ be three disjoint graphs, ${P}_{3}={u}_{1}{u}_{2}{u}_{3}$ with ${d}_{{P}_{3}}\left({u}_{2}\right)=2$ and ${v}_{j}\in V\left({G}_{j}\right)$ for each $j\in \mathcal{S}$${}_{3}$. Define ${P}_{3}\circ ({G}_{1},{G}_{2},{G}_{3})$ as the graph obtained from the disjoint union ${G}_{1},{G}_{2},{G}_{3}$ and ${P}_{3}$ by identifying ${u}_{j}$ with ${v}_{j}$ (again denoted as ${v}_{j}$) for each $j\in \mathcal{S}$${}_{3}$.
- (v)
- For $i\in \{1,2,\dots ,t\}$ with $t\ge 3$, let ${G}_{i}$ be a graph with ${D}_{1}\left({G}_{i}\right)\ne \varnothing $ and ${z}_{i}\in {D}_{1}\left({G}_{i}\right)$. Define $G={\oplus}_{{z}_{0}}({G}_{1},{G}_{2},...,{G}_{t})$ to be a graph obtained from disjoint union of ${G}_{1},{G}_{2},\dots ,{G}_{t}$ by identifying ${z}_{1},{z}_{2},\dots ,{z}_{t}$ into a single vertex ${z}_{0}$ in G. As ${z}_{0}={z}_{i}$ in ${G}_{i}$, ${z}_{0}$ is viewed as the vertex ${z}_{i}$ in ${G}_{i}$.
- (vi)
- If $\left|V\right(G\left)\right|\ge 3$, then define $\mathcal{M}$$\left(G\right)=\{G-uv:uv\in E\left(G\right)\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}uv\phantom{\rule{4pt}{0ex}}is\phantom{\rule{4pt}{0ex}}not\phantom{\rule{4pt}{0ex}}a\phantom{\rule{4pt}{0ex}}cut\phantom{\rule{4pt}{0ex}}edge\}\cup \phantom{\rule{4pt}{0ex}}\{G-v:v\in {D}_{1}\left(G\right)\}$ to be the family of all proper maximal subgraphs of G.

**Definition**

**2.**

**Theorem**

**1**

**Corollary**

**1.**

**Lemma**

**5**

**Theorem**

**2**

**Corollary**

**2**

**Theorem**

**3.**

**Proof.**

**Corollary**

**3.**

- (1)
- ${G}_{1},{G}_{3}$ are 2-connected;
- (2)
- ${v}_{j}$ is a small-cut vertex corresponding to an optimal labeling ${\pi}_{j}$ of ${G}_{j}$ for each $j\in \mathcal{S}$${}_{3}$;
- (3)
- ${G}_{1},{G}_{2}$, ${G}_{3}$ are $(k-1)$-cutwidth critical, then ${P}_{3}\circ ({G}_{1},{G}_{2},{G}_{3})$ is k-cutwidth critical, where ${G}_{1},{G}_{2},{G}_{3}$ are not necessarily distinct.

**Proof.**

**Proof.**

- (1)
- $\{{H}_{i}:1\le i\le q\}$, or
- (2)
- $\{{H}_{i}^{\prime}:1\le i<q\}$ in which ${H}_{i}^{\prime}$ may be one of $\{{H}_{i},{H}_{i-1}\cup {H}_{i}\cup {H}_{i+1}\}$ with ${H}_{0}={H}_{q},{H}_{q+1}={H}_{1}$, and there exists at least ${H}_{i}^{\prime}\ne {H}_{i}$, or
- (3)
- $\{{H}_{i}^{\u2033}:1\le i<q\}$, each of which is either ${H}_{i}$ or ${H}_{i-1}\left[{E}^{\prime}\right]\cup {H}_{i}\cup {H}_{i+1}\left[{E}^{\u2033}\right]$ with ${H}_{0}={H}_{q},{H}_{q+1}={H}_{1}$, and there exists at least ${H}_{i}^{\u2033}\ne {H}_{i}$, where ${H}_{i-1}\left[{E}^{\prime}\right]\subset {H}_{i-1}$ and ${H}_{i+1}\left[{E}^{\u2033}\right]\subset {H}_{i+1}$.

**Theorem**

**4.**

- (1)
- G has a central vertex ${v}_{0}$, and ${v}_{0}$-components of $G-{v}_{0}$ constitute a decomposition $\mathcal{S}$ with $\left|\mathcal{S}\right|$$=3$, each of which is ${K}_{2}$ with cutwidth 1;
- (2)
- G is a cycle ${C}_{3}$, whose three edges constitute a decomposition $\mathcal{S}$ with $\left|\mathcal{S}\right|$$=3$, each element of which is ${K}_{2}$ with cutwidth 1.

**Theorem**

**5.**

- (1)
- has a central vertex ${v}_{0}$, and ${v}_{0}$-components of $G-{v}_{0}$ constitute a decomposition $\mathcal{S}$ with $\left|\mathcal{S}\right|$$=3$, each of which equals ${K}_{1,3}$ or ${C}_{3}$ with cutwidth 2; or
- (2)
- G has a central cycle ${C}_{3}={x}_{1}{x}_{2}{x}_{3}{x}_{1}$ with ${d}_{G}\left({x}_{i}\right)$ = 3 for ${x}_{i}\in V\left({C}_{3}\right)$, and ${C}_{3}$-components of $G-E\left({C}_{3}\right)$ constitute a decomposition $\mathcal{S}$ with $\left|\mathcal{S}\right|$$=3$, each member of which equals ${K}_{1,3}$ with cutwidth 2; or
- (3)
- G equals ${C}_{4}+{x}_{1}{x}_{3}$ or ${C}_{4}+{x}_{2}{x}_{4}$, where ${C}_{4}={x}_{1}{x}_{2}{x}_{3}{x}_{4}{x}_{1}$ is a cycle of length 4.

## 3. 4-Cutwidth Critical Graphs with a Central Vertex

#### 3.1. 4-Cutwidth Critical Trees with a Central Vertex

**Definition**

**3.**

**Theorem**

**6.**

- (1)
- T possesses a configuration ${K}_{1,3}\circ ({T}_{1},{T}_{2},{T}_{3})$ which can be decomposed into three edge-disjoint 3-cutwidth trees ${T}_{1},{T}_{2}$ and ${T}_{3}$ (not necessarily distinct), and the 3-degree vertex of ${K}_{1,3}$ is the central vertex of T, where ${T}_{i}$ is a ${v}_{0}$-component of $T-{v}_{0}$ with either ${T}_{i}\in \{{\tau}_{1},{\tau}_{2}\}$ or ${T}_{i}-{v}_{0}\in \{{\tau}_{1},{\tau}_{2}\}$ for each $1\le i\le 3$ (see Figure 3d); or
- (2)
- T is a tree with a central vertex ${v}_{0}$ with ${d}_{T}\left({v}_{0}\right)\ge 4$ and with an edge-joint decomposition $\{{T}_{1},{T}_{2},{T}_{3}\}$ of equal cutwidth 3, where ${T}_{1},{T}_{2}$ and ${T}_{3}$ (not necessarily distinct), which are defined by (7), are either in $\{{\tau}_{1},{\tau}_{2}\}$ or homeomorphic to ${\tau}_{2}$, and at least one of them, say ${T}_{3}$, is not ${\tau}_{1}$ (see Figure 3a–c, respectively).

#### 3.2. 4-Cutwidth Critical Nontrees with a Central Vertex

**Lemma**

**7.**

**Proof.**

**Corollary**

**4.**

**Corollary**

**5.**

**Lemma**

**8.**

- (i)
- each non cut-edge of ${G}_{2}$ may be subdivided once, and ${v}_{2}$ may possibly be the subdivision vertex;
- (ii)
- ${G}_{2}\ne {\tau}_{1}$;
- (iii)
- if ${G}_{2}\in \{{\tau}_{2},{\tau}_{3}\}$, then ${v}_{2}$ is not either the central vertex or the pendant vertex of it;
- (iv)
- ${G}_{i}\notin \{{\tau}_{2},{\tau}_{3}\}$ for $i=1$ or 3 if ${G}_{2}\in \{{\tau}_{2},{\tau}_{3}\}$.

**Proof.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

- (1)
- ${G}_{1},{G}_{2}$ are in $\{{\tau}_{i}:2\le i\le 5\}$;
- (2)
- at least one of ${G}_{1}$ and ${G}_{2}$, say ${G}_{1}$, is in $\{{\tau}_{2},{\tau}_{3}\}$, while ${G}_{2}\ne {\tau}_{2}$;
- (3)
- ${v}_{0}$ is the central vertex of ${G}_{1}$, but ${v}_{0}$ is only a vertex of any 3-cycle ${C}_{3}$ of ${G}_{2}$.

**Proof.**

**Definition**

**4.**

**Lemma**

**11.**

**Proof.**

**Lemma**

**12.**

**Proof.**

**Lemma**

**13.**

**Proof.**

- (1)
- ${\overline{G}}_{1}={K}_{1,3}$, one of whose three pendant vertices is ${v}_{0}$, ${\overline{G}}_{2}\in \{{\tau}_{2},{\tau}_{3}\}$ and ${\overline{G}}_{3}\in \{{\tau}_{3},{\tau}_{4},{\tau}_{5}\}$;
- (2)
- ${\overline{G}}_{1}={C}_{3}$, one of whose three 2-degree vertices is ${v}_{0}$, ${\overline{G}}_{2}\in \{{\tau}_{2},{\tau}_{3}\}$ and ${\overline{G}}_{3}\in \{{\tau}_{3},{\tau}_{4},{\tau}_{5}\}$.

- (a1)
- ${d}_{G}\left({v}_{0}\right)\ge 7$ because of $c\left({K}_{1,7}\right)=4$;
- (a2)
- ${M}_{2}$ is a subgraph of G because of $c\left({M}_{2}\right)=4$;
- (a3)
- G is a tree because G is a non-tree graph;
- (a4)
- $\{{\overline{G}}_{1},{\overline{G}}_{2},{\overline{G}}_{3}\}$ is a decomposition of equal cutwidth 3;
- (a5)
- $c\left(G\right)=3$ because G is 4-cutwidth critical.

**Theorem**

**7.**

- (1)
- For $1\le i\le 3$, if ${G}_{i}$ is some ${\tau}_{i}$$(1\le i\le 5)$ in Figure 1 and ${G}_{i}^{\prime}$ corresponding to ${G}_{i}$ is a graph defined in $\left(8\right)$, then $G={K}_{1,3}\circ ({G}_{1}^{\prime},{G}_{2}^{\prime},{G}_{3}^{\prime})$, where ${G}_{1},{G}_{2}$ and ${G}_{3}$ are not necessarily different;
- (2)
- $G={P}_{3}\circ ({G}_{1}^{\prime},{G}_{2},{G}_{3}^{\prime})$, where ${G}_{i}\in \{{\tau}_{i}:1\le i\le 5\}$ with ${v}_{i}\in V\left({G}_{i}\right)$ for $1\le i\le 3$ and ${G}_{i}^{\prime}$ corresponding to ${G}_{i}$ is a graph defined in $\left(8\right)$, ${G}_{i}\notin \{{\tau}_{2},{\tau}_{3}\}$ for $i=1,3$ and ${G}_{i}\ne {\tau}_{1}$ for $i=2$ ${v}_{2}$ is not either the central vertex or the pendent vertex when ${G}_{2}\in \{{\tau}_{2},{\tau}_{3}\}$ but ${v}_{2}$ is possible to a subdivision vertex of a non cut-edge of ${G}_{2}$ when ${G}_{2}\in \{{\tau}_{3},{\tau}_{4},{\tau}_{5}\}$;
- (3)
- $G={G}_{1}{\oplus}_{{u}_{1},{x}_{1}}{G}_{2}^{\prime}$ with the central vertex ${u}_{1}$ of ${d}_{G}\left({u}_{1}\right)<7$, where ${G}_{1}\in \{{\tau}_{2},{\tau}_{3}\}$ with the central vertex ${u}_{1}$$({u}_{1}={v}_{0}$ of ${\tau}_{2}$ or ${\tau}_{3}$, respectively, see Figure 1$),$ ${G}_{2}\in \{{\tau}_{3},{\tau}_{4},{\tau}_{5}\}$ with a 3-cycle ${C}_{3}\subset {G}_{2}$ and ${x}_{1}\in V\left({C}_{3}\right)$ with ${d}_{{G}_{1}}\left({u}_{1}\right)+{d}_{{G}_{2}}\left({x}_{1}\right)\le 6$, ${G}_{2}^{\prime}$ corresponding to ${G}_{2}$ is a graph defined in $\left(8\right)$;
- (4)
- G has a subgraph decomposition $\{{\overline{G}}_{1},{\overline{G}}_{2},{\overline{G}}_{3}\}$ of equal cutwidth 3, defined in Definition 4, where G is a graph with a central vertex ${v}_{0}$ of ${d}_{G}\left({v}_{0}\right)\ge 4$ and at least two cut edges ${v}_{0}{v}_{1},{v}_{0}{v}_{2}$, ${\overline{G}}_{i}$ is 3-cutwidth critical for $1\le i\le 3$;
- (5)
- G has a subgraph decomposition $\{{C}_{3},{C}_{3}^{\prime},{C}_{3}^{\u2033}\}$ of equal cutwidth 2, each of which is a ${v}_{0}$-component of $G-{v}_{0}$, where ${v}_{0}$ is the central vertex ${v}_{0}$ of degree 6 of G, and ${C}_{3}^{\prime}$ and ${C}_{3}^{\u2033}$ are the copies of a 3-cycle ${C}_{3}$;
- (6)
- G is one member of $\{{M}_{i}:9\le i\le 17\}$ with a central vertex ${v}_{0}$ (see Figure 4) and a subgraph decomposition $\{{\overline{G}}_{1},{\overline{G}}_{2},{\overline{G}}_{3}\}$, in which ${\overline{G}}_{1}={K}_{1,3}$, one of whose pendant vertices is ${v}_{0}$, ${\overline{G}}_{i}\in \{{\tau}_{3},{\tau}_{4},{\tau}_{5}\}$ for $i=2,3$, where ${\overline{G}}_{i}$ satisfies:

- (i)
- ${v}_{0}$ is a 2-degree vertex y of ${C}_{3}$ of ${\overline{G}}_{i}$ for ${\overline{G}}_{i}={\tau}_{3}$;
- (ii)
- if the 3-degree vertex of ${\overline{G}}_{1}(={K}_{1,3})$ is x and ${\overline{G}}_{i}={\tau}_{4}$, then ${\tau}_{4}={H}_{2}+{v}_{0}x$ and ${v}_{0}$ is a 3-degree vertex of ${\overline{G}}_{i}$;
- (iii)
- ${v}_{0}$ is either a 2-degree vertex of ${\overline{G}}_{i}$ or a 3-degree vertex of ${\overline{G}}_{i}$ for ${\overline{G}}_{i}={\tau}_{5}$, but if ${\overline{G}}_{2}={\overline{G}}_{3}={\tau}_{5}$ and ${v}_{0}$ is a 3-degree vertex of ${\overline{G}}_{2}$, then ${v}_{0}$ must not be a 3-degree vertex of ${\overline{G}}_{3}$, and vice versa.

## 4. 4-Cutwidth Critical Graphs with a Central Cycle

**Lemma**

**14.**

**Proof.**

#### 4.1. Graphs with a Central Cycle of Length Three

**Definition**

**5.**

**Lemma**

**15.**

**Proof.**

**Lemma**

**16.**

**Proof.**

**Lemma**

**17.**

**Proof.**

- (1)
- ${H}_{1}={G}_{1}\cup G\left[\{{x}_{1}{x}_{2},{x}_{1}{x}_{3}\}\right]$ with ${d}_{G}\left({x}_{1}\right)\ge 4$, ${H}_{2}={G}_{2}$ and ${H}_{3}={G}_{3}$;
- (2)
- ${H}_{1}={G}_{1}\cup G\left[\{{x}_{1}{x}_{2},{x}_{1}{x}_{3}\}\right]$ with ${d}_{G}\left({x}_{1}\right)\ge 4$, ${H}_{2}={G}_{2}\cup G\left[\{{x}_{2}{x}_{1},{x}_{2}{x}_{3}\}\right]$ and ${H}_{3}={G}_{3}$;
- (3)
- ${H}_{1}={G}_{1}\cup G\left[\{{x}_{1}{x}_{2},{x}_{1}{x}_{3}\}\right]$ with ${d}_{G}\left({x}_{1}\right)\ge 4$, ${H}_{2}={G}_{2}\cup G\left[\{{x}_{2}{x}_{1},{x}_{2}{x}_{3}\}\right]$ and ${H}_{3}={G}_{3}\cup G\left[\{{x}_{3}{x}_{1},{x}_{3}{x}_{2}\}\right]$;
- (4)
- ${H}_{1}={G}_{1}\cup {C}_{3}$ with ${d}_{G}\left({x}_{1}\right)\ge 4$, ${H}_{2}={G}_{2}$ and ${H}_{3}={G}_{3}$;
- (5)
- ${H}_{1}={G}_{1}\cup {C}_{3}$ with ${d}_{G}\left({x}_{1}\right)\ge 4$, ${H}_{2}={G}_{2}\cup G\left[\{{x}_{2}{x}_{1},{x}_{2}{x}_{3}\}\right]$ and ${H}_{3}={G}_{3}$;
- (6)
- ${H}_{1}={G}_{1}\cup {C}_{3}$ with ${d}_{G}\left({x}_{1}\right)\ge 4$, ${H}_{2}={G}_{2}\cup G\left[\{{x}_{2}{x}_{1},{x}_{2}{x}_{3}\}\right]$ and ${H}_{3}=$${G}_{3}\cup G\left[\{{x}_{3}{x}_{1},{x}_{3}{x}_{2}\}\right]$;
- (7)
- ${H}_{1}={G}_{1}\cup G\left[\{{x}_{1}{x}_{2},{x}_{1}{x}_{3}\}\right]$, ${H}_{2}={G}_{2}$ and ${H}_{3}={G}_{3}$;
- (8)
- ${H}_{1}={G}_{1}\cup G\left[\{{x}_{1}{x}_{2},{x}_{1}{x}_{3}\}\right]$, ${H}_{2}={G}_{2}\cup G\left[\{{x}_{2}{x}_{1},{x}_{2}{x}_{3}\}\right]$ and ${H}_{3}={G}_{3}$;
- (9)
- ${H}_{1}={G}_{1}\cup G\left[\{{x}_{1}{x}_{2},{x}_{1}{x}_{3}\}\right]$, ${H}_{2}={G}_{2}\cup G\left[\{{x}_{2}{x}_{1},{x}_{2}{x}_{3}\}\right]$ and ${H}_{3}=$${G}_{3}\cup G\left[\{{x}_{3}{x}_{1},{x}_{3}{x}_{2}\}\right]$,

**Lemma**

**18.**

**Proof.**

**Lemma**

**19.**

**Proof.**

**Lemma**

**20.**

**Proof.**

**Theorem**

**8.**

- (1)
- G has a decomposition $\{{H}_{1},{H}_{2},{H}_{3}\}$ of nonequal cutwidth ρ with ρ=2 or 3, each of which is ρ-cutwidth critical, where ${x}_{i}$ is a cut vertex for each $1\le i\le 3$ and there are at least two vertices (say ${x}_{2},{x}_{3}$) such that ${d}_{G}\left({x}_{2}\right)\ge 4$ and ${d}_{G}\left({x}_{3}\right)\ge 4$ (see ${M}_{5}$–${M}_{7}$ in Figure 2 and Illustration in Figure 6d,e);
- (2)
- G has a decomposition $\{{H}_{1},{H}_{2},{H}_{3}\}$ of equal cutwidth 3 in which ${H}_{i}$ or ${H}_{i}-{x}_{i}{x}_{i}^{\prime}$ with ${x}_{i}^{\prime}\in {N}_{G}\left({x}_{i}\right)\cap V\left({G}_{i}\right)$ is 3-cutwidth critical, and at least a ${H}_{i}$ (say ${H}_{1}$) contains at least two edges ${x}_{1}{x}_{2}$ and ${x}_{1}{x}_{3}$ of ${C}_{3}$, where ${x}_{i}$ is a cut vertex for each $1\le i\le 3$ and there is at most a vertex (say ${x}_{3}$) such that ${d}_{G}\left({x}_{3}\right)\ge 4$ (see Illustration in Figure 6a–c);
- (3)
- G is 2-connected and $G={M}_{8}$ (see Figure 2) with a decomposition $\{{H}_{1},{H}_{2},{H}_{3}\}$ of equal cutwidth 3 in which ${H}_{i}=G\left[\{{x}_{1},{x}_{2},{x}_{3},{y}_{i}\}\right]={\tau}_{5}$ for $1\le i\le 3$;
- (4)
- $G\in \{{M}_{3},{M}_{4},{M}_{5},{M}_{6},{M}_{7}\}$ with an edge-disjoint decomposition $\{{G}_{1},{G}_{2},{G}_{3},{C}_{3}\}$ of equal cutwidth 2, in which ${G}_{i}$ is either ${K}_{1,3}$ or a copy ${C}_{3}^{\prime}$ of ${C}_{3}$ for $1\le i\le 3$ (see ${M}_{3}$–${M}_{7}$ in Figure 2).

#### 4.2. Graphs with a Central Cycle of Length Four

**Lemma**

**21.**

**Proof.**

**Lemma**

**22.**

**Proof.**

**Theorem**

**9.**

- (1)
- ${\overline{G}}_{1}={K}_{1,5}$ with the central vertex ${x}_{1}$ of ${d}_{G}\left({x}_{1}\right)=5$ or ${\tau}_{5}$ with ${d}_{G}\left({x}_{1}\right)=4$, and ${\overline{G}}_{2}$ and ${\overline{G}}_{4}$ are both in $\{{\tau}_{2},{\tau}_{3}\}$, but ${\overline{G}}_{2}$ and ${\overline{G}}_{4}$ do not equal ${\tau}_{3}$ simultaneously (see Illustration in Figure 7a);
- (2)
- ${\overline{G}}_{1}$ is homeomorphic to ${\tau}_{3}$ with the central vertex ${x}_{1}$ of ${d}_{G}\left({x}_{1}\right)=4$ and ${C}_{4}\subset {\overline{G}}_{1}$, ${\overline{G}}_{2}$ and ${\overline{G}}_{4}$ are both in $\{{\tau}_{2},{\tau}_{3}\}$. ${\overline{G}}_{2},{\overline{G}}_{4}$ are not necessarily different (see Illustration in Figure 7b).

#### 4.3. Graphs with a Central Cycle of Length at Least Five

- (1)
- ${\overline{G}}_{1}={G}_{1}\cup {G}_{2}\cup {G}_{5}\cup ({C}_{5}-{x}_{3}{x}_{4})$, ${\overline{G}}_{i}={G}_{i}$ or ${G}_{i}+{x}_{3}{x}_{4}$ if $c\left({G}_{i}\right)=3$ or ${G}_{i}+{x}_{i}{x}_{i-1}+{x}_{i}{x}_{i+1}$ if $c\left({G}_{i}\right)<3$ for $i=3,4$ with ${d}_{G}\left({x}_{3}\right)={d}_{G}\left({x}_{4}\right)=3$;
- (2)
- ${\overline{G}}_{1}={G}_{1}\cup {G}_{2}\cup {G}_{5}\cup ({C}_{5}-{x}_{3}{x}_{4})$, ${\overline{G}}_{3}={G}_{3}\cup ({C}_{5}-{x}_{1}{x}_{5}+{x}_{2}{x}_{2}^{\prime}+{x}_{4}{x}_{4}^{\prime})$, ${\overline{G}}_{4}={G}_{4}\cup ({C}_{5}-{x}_{1}{x}_{2}+{x}_{3}{x}_{3}^{\prime}+{x}_{5}{x}_{5}^{\prime})$ with ${d}_{G}\left({x}_{3}\right)={d}_{G}\left({x}_{4}\right)=4$ and $c\left({G}_{3}\right)=c\left({G}_{4}\right)=2$, ${x}_{i}^{\prime}\in {N}_{G}\left({x}_{i}\right)\cap V({G}_{i}-{x}_{i})$ for $2\le i\le 5$;
- (3)
- ${\overline{G}}_{1}$ is homeomorphic to subgraph $({G}_{1}+{x}_{1}{x}_{2}+{x}_{1}{x}_{5})\cup {G}_{2}\cup {G}_{5}$, ${\overline{G}}_{3}={G}_{3}\cup ({C}_{5}-{x}_{1}{x}_{5}+{x}_{2}{x}_{2}^{\prime}+{x}_{4}{x}_{4}^{\prime})$, ${\overline{G}}_{4}={G}_{4}\cup ({C}_{5}-{x}_{1}{x}_{2}+{x}_{3}{x}_{3}^{\prime}+{x}_{5}{x}_{5}^{\prime})$ with $c\left({G}_{3}\right)=c\left({G}_{4}\right)=2$, where ${C}_{5}$ has at most two 4-degree vertices (say, ${x}_{1}$ and ${x}_{4}$) which are nonadjacent.

**Lemma**

**23.**

**Proof.**

**Lemma**

**24.**

**Proof.**

**Theorem**

**10.**

- (1)
- ${\overline{G}}_{1}\in \{{\tau}_{2},{\tau}_{3}\}$ with the central vertex ${x}_{1}$ of degree three or four, for $i=3,4$, ${\overline{G}}_{i}$(or ${\overline{G}}_{i}-{x}_{i})$ is one of $\{{\tau}_{i}:1\le i\le 5\}$ and ${x}_{i}$ satisfies: $\left(i\right)$ ${d}_{G}\left({x}_{i}\right)=3$, $\left(ii\right)$ ${x}_{i}$ is not the central vertex of ${\overline{G}}_{i}$ when ${\overline{G}}_{i}\in \{{\tau}_{1},{\tau}_{2},{\tau}_{3}\}$, and $\left(iii\right)$ ${x}_{i}{x}_{i-1},{x}_{i}{x}_{i+1}$ are the pendant edges of ${\overline{G}}_{i}$ when ${\overline{G}}_{i}$ is ${\tau}_{2}$ or ${\tau}_{3}$ (see Illustration in Figure 8a);
- (2)
- ${\overline{G}}_{1}$ is homeomorphic to ${\tau}_{2}$ with the central vertex ${x}_{1}$ of degree three, for $i=3,4$, ${\overline{G}}_{i}$ is homeomorphic to ${\tau}_{2}$ or ${\tau}_{3}$ with ${G}_{i}\in \{{K}_{1,3},{C}_{3}\}$, where ${G}_{3},{G}_{4}$ are not necessarily different (see Illustration in Figure 8b);
- (3)
- ${\overline{G}}_{1}$ is homeomorphic to ${\tau}_{3}$ with the central vertex ${x}_{1}$ of degree four, for $i=3,4$, ${\overline{G}}_{i}$ is homeomorphic to ${\tau}_{2}$ or ${\tau}_{3}$ with ${G}_{i}\in \{{K}_{1,3},{C}_{3}\}$, but if ${G}_{3}={C}_{3}$, then ${G}_{4}\ne {C}_{3}$ and vice versa (see Illustration in Figure 8b).

**Lemma**

**25.**

**Proof.**

**Theorem**

**11.**

## 5. 4-Cutwidth Critical Graphs without a Central Vertex and Central Cycle

**Theorem**

**12.**

**Proof.**

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Zhang, Z.; Lai, H. Structures of Critical Nontree Graphs with Cutwidth Four. *Mathematics* **2023**, *11*, 1631.
https://doi.org/10.3390/math11071631

**AMA Style**

Zhang Z, Lai H. Structures of Critical Nontree Graphs with Cutwidth Four. *Mathematics*. 2023; 11(7):1631.
https://doi.org/10.3390/math11071631

**Chicago/Turabian Style**

Zhang, Zhenkun, and Hongjian Lai. 2023. "Structures of Critical Nontree Graphs with Cutwidth Four" *Mathematics* 11, no. 7: 1631.
https://doi.org/10.3390/math11071631