Graph burning is a model that describes the spread of social contagion on social networks such as Facebook or Twitter. We use Bondy and Murty [1] for the notation and terminology, burning process is defined as follows. Given a finite and simple graph $G$, vertices may be either burned or unburned throughout the process. Initially, at time $t=0$, all vertices are unburned. At each time $t\ge 1$, one new unburned vertex is chosen to burn, if such a vertex is available, we call such a chosen vertex a fire source. If a vertex is burned, then it remains in that state until the end of the process. Once a vertex is burned at time $t$, at time $t+1$ each of its unburned neighbors becomes burned. The process ends when all vertices of $G$ are burned.

Note that the burning process on $G$ may be highly dependent on the choice of fire sources, the strategic choice of sources is critical when minimizing the length of the process. In [2] , Bonato introduced the burning number of graph $G$, denoted by $b\left(G\right)$, is the minimum steps to burn graph $G$. The fire sources $x_1,\cdots ,x_k$ that are chosen over time on graph $G$ are referred to as a burning sequence $\left(x_1,\cdots ,x_k\right)$ of $G$ and call the shortest burning sequence optimal. Clearly, optimal burning sequences have length $b\left(G\right)$.

In 2016, Bonato et al proposed the burning conjecture:

Conjecture 1.1 [3] . For a connected graph $G$ of order $n$, $b\left(G\right)\le \lceil \sqrt{n}\rceil$.

Later, the conjecture been comfirm for some graph classes, such as spiders [4] , path forests [5] , caterpillars [6] [7] , theta graph [8] , fence graph [9] , generalized Petersen graph [10] , the Cartesian product of paths [11] , $Q$ graph [12] , binary tree [13] and trees without degree-2 vertices [14] .

Motivated by these, we put forward on the paths and circles. We first denote octopus graph $G$ which obtained from comet graph $C_{l_i,2}$ for $1\le i\le m$ by identifing the tail of $C_{l_i,2}$ at $v_0$, clearly, $d\left(v_0\right)=m$. All degree-3 vertices but $v_0$ denoted by $u_i$, we call $C_{l_i,2}$ an arm of Octopus and $l_i$ is the length of arm $C_{l_i,2}$ (see Figure 1 ). Here the comet $C_{r,s}$ is a graph which obtained by the end of path $P_r$ with the center of star graph $K_{1,s}$.

Another class of graph named $t$ tail bicyclic graph. If $t=1$, call single tail bicyclic graph which obtained by joining the center vertex of the bicyclic graphs with one vertex of the path $P_{a_1+1}$ denoted by $G_{v_0\left(g_1,g_2,a_1\right)}$. If $t=2$, call double tail bicyclic graph which obtained by joining the center vertex of the bicyclic graph with a vertex of path $P_{a_1+1}$ and path $P_{a_2+1}$ denoted by $G_{v_0\left(g_1,g_2,a_1,a_2\right)}$ (see Figure 2 ). They all only have a vertex $v_0$ with $d\left(v_0\right)>2$, without loss of generality, we suppose $g_1\ge g_2$.

In this paper, we first confirm the burning conjecture for these 3 kind of graphs, next we discuss the burning number of single tail bicyclic graph $G_{v_0\left(g_1,g_2,a_1\right)}$ and double tail bicyclic graph $G_{v_0\left(g_1,g_2,a_1,a_2\right)}$.

Lemma 2.1 [2] If $\left(x_1,x_2,\cdots ,x_k\right)$ is a sequence of nodes in a graph $G$ such that $N_{k-1}\left[x_1\right]\cup N_{k-2}\left[x_2\right]\cup \cdots \cup N_1\left[x_{k-1}\right]\cup N_0\left[x_k\right]=V\left(G\right)$, then $b\left(G\right)\le k$.

Lemma 2.2 [3] For a path $P_n$ or a cycle $C_n$ on $n$ nodes, we have $b\left(P_n\right)=b\left(C_n\right)\le \lceil \sqrt{n}\rceil$.

Lemma 2.3 [3] For a graph $G$, $b\left(G\right)=\min \left\{b\left(T\right):T\text{is}\text{a}\text{spanning}\text{tree}\text{of}G\right\}$.

Lemma 2.4 [3] For any graph $G$ with radius $r$ and diameter $d$, we have that $\lceil {\left(d+1\right)}^{\frac{1}{2}}\rceil \le b\left(G\right)\le r+1$.

Lemma 2.5 [4] The burning number of a spider graph G of order n satisfies $b\left(G\right)\le \lceil \sqrt{n}\rceil$.

Lemma 2.6 [4] If $G$ is a path-forest of order $n$ with $t\ge 1$ components, then $b\left(G\right)\le \lfloor \frac{n}{2}\rfloor +t$.

Lemma 2.7 [5] Let $G=P_{a_1}\cup P_{a_2}$ with $a_1\ge a_2\ge 1$ and $J\left(t\right)=\left\{\left(t^2-2,2\right)\right\}$ for integer $t\ge 2$. Then

$b\left(G\right)=\{\begin{array}{ll}\lceil \sqrt{a_1+a_2}\rceil +1,\hfill & \text{If}\left(a_1,a_2\right)\in J\left(t\right);\hfill \\ \lceil \sqrt{a_1+a_2}\rceil ,\hfill & \text{Otherwise}.\hfill \end{array}$

Lemma 2.8 [5] Let $G=P_{a_1}\cup P_{a_2}\cup P_{a_3}$ with $a_1\ge a_2\ge a_3\ge 1$. Then

$b\left(G\right)=\{\begin{array}{ll}\lceil \sqrt{a_1+a_2+a_3}\rceil +1,\hfill & \text{If}\left(a_1,a_2,a_3\right)\in J^1\cup J^2\cup J^3\cup J^4\cup J^5;\hfill \\ \lceil \sqrt{a_1+a_2+a_3}\rceil ,\hfill & \text{Otherwise}.\hfill \end{array}$

Let $J^i$ for $1\le i\le 5$ satisfy the following conditions.

$D_1=\left\{\left(2,2\right)\right\},$

$D_2=\left\{\left(3,2\right)\right\},$

$D_3=\left\{\left(1,1\right),\left(3,3\right),\left(4,2\right),\left(5,5\right)\right\},$

$D_4=\left\{\left(2,1\right),\left(4,1\right),\left(4,3\right),\left(4,4\right),\left(6,1\right),\left(6,4\right),\left(6,5\right),\left(6,6\right),\left(7,7\right),\left(8,4\right),\left(8,6\right),\left(10,4\right)\right\},$

$\begin{array}{l}{D}_5=\{\left(11,10,4\right),\left(13,11,1\right),\left(11,11,3\right),\left(22,13,1\right),\left(19,13,4\right),\left(17,13,6\right),\left(15,13,8\right),\\ \left(13,13,10\right),\left(17,15,4\right),\left(15,15,6\right),\left(30,15,4\right),\left(28,15,6\right),\left(26,15,8\right),\left(19,15,15\right),\\ \left(28,17,4\right),\left(26,17,6\right),\left(17,17,15\right),\left(26,19,4\right),\left(43,17,4\right),\left(41,17,6\right),\left(30,17,17\right),\\ \left(41,19,4\right),\left(30,30,4\right),\left(58,19,4\right)\},\end{array}$

$J^1=\left\{\left(a_1,a_2,a_3\right):\left(a_2,a_3\right)\in D_1,a_1+a_2+a_3=t^2-3\text{for}\text{integert}\right\},$

$J^2=\left\{\left(a_1,a_2,a_3\right):\left(a_2,a_3\right)\in D_1\cup D_2,a_1+a_2+a_3=t^2-2\text{for}\text{integert}\right\},$

$J^3=\left\{\left(a_1,a_2,a_3\right):\left(a_2,a_3\right)\in {\displaystyle \underset{i=1}{\overset{3}{\cup }}{D}_i},a_1+a_2+a_3=t^2-1\text{for}\text{integert}\right\},$

$J^4=\left\{\left(a_1,a_2,a_3\right):a_3=2\text{or}\left(a_2,a_3\right)\in {\displaystyle \underset{i=1}{\overset{4}{\cup }}{D}_i},a_1+a_2+a_3=t^2\text{for}\text{integert}\right\},$

$J^5=D_5\cup \left\{11,11,2\right\}.$

In this section, we first confirm the burning conjecture for octopus graph $G$, single tail bicyclic graph $G_{v_0\left(g_1,g_2,a_1\right)}$ and double tail bicyclic graph $G_{v_0\left(g_1,g_2,a_1,a_2\right)}$.

Theorem 3.1 Let $G$ be a octopus graph with order $n$. Then $b\left(G\right)\le \lceil \sqrt{n}\rceil$

Proof. Let $G$ be a octopus graph with order $n$. Without loss of generality, suppose $n=q^2+p$ for $1\le p\le 2q+1$. The neighbors of $v_0$ are $v_1,v_2,\cdots ,v_i$ respectively and the length of longest arm is $l$. If $q=2$, it’s clearly that $b\left(G\right)\le \lceil \sqrt{n}\rceil$. Consider $q\ge 3$, next we distinguish 3 cases to complete the proof.

Case 1 If $l\le q$.

It’s clearly that radius of $G$ is $r\le q$, by lemma 2.4, we have $b\left(G\right)\le \lceil \sqrt{n}\rceil$.

Case 2 If $q+1\le l\le 2q-1$.

Consider the structure of the arm, we will discuss two cases.

Subcase 2.1 The arms of $G$ have the same structure.

If each arm has the same length, then $G$ has $q$ arms of length $q+1$. First we set the $x_1$ on $v_1$, then $C_{l_1,2}\subseteq N_q\left[x_1\right]$. We denoted the part of $G\backslash N_q\left[x_1\right]$ is $R=R_1\cup R_2\cup \cdots \cup R_{q-1}$, clearly, the height of $R_i\left(1\le i\le q-1\right)$ is 2 and each $R_i$ contains $u_i$. Now suppose $\left(x_2,x_3,\cdots ,x_q\right)$ is a burning sequence of $R$ and let $x_{i+1}=u_i$ for $1\le i\le q-1$, then we have $V\left(G\right)\subseteq N_q\left[x_1\right]\cup N_{q-1}\left[x_2\right]\cup N_{q-2}\left[x_3\right]\cup \cdots \cup N_1\left[x_q\right]$. By lemma 2.1, we have $b\left(G\right)\le \lceil \sqrt{n}\rceil =q+1$.

Subcase 2.2 The arms of $G$ doesn’t have the same structure.

It’s clearly that $G$ doesn’t have $q$ arms of length $q+1$. We set $x_1$ on $v_0$, if $l_i\le q$, then $C_{l_i,2}\subseteq N_q\left[x_1\right]$. We denoted the part of $G\backslash N_q\left[x_1\right]$ is $R_1\cup R_2\cup \cdots \cup R_i\left(i\ge 1\right)$. Since the arm of octopus $G$ has length at most $2q-1$, then each $R_i$ has length and order at most $q-2$ and $q$ respectively. Next we discuss the burning number of $G$ by the number of $i$ compontent.

When $i\le \frac{q+1}{2}$. We remove a pendant vertex to another pendant vertex such that $R_i$ become a path $P_i$, $P=P_1\cup P_2\cup \cdots \cup P_i\left(i\ge 1\right)$, it’s clearly that $b\left(R\right)\le b\left(P\right)$. Next we denoted $w_k$ is ssthe center of $P_k$, for each $k=1,2,\cdots ,i$, we can by neighborhood $N_{q-k}\left[w_k\right]$ cover the $P_k$. since $2\left(q-k\right)+1\ge 2\cdot \frac{q+1}{2}+1=q$, thus $b\left(R\right)\le b\left(P\right)\le q$. By lemma 2.1, we have $b\left(G\right)\le \lceil \sqrt{n}\rceil =q+1$.

When $i=\frac{q+2}{2}$, $q$ is even. For each $k=1,2,\cdots ,i-1$, every $R_k$ can be covered by $N_{q-k}\left[w_k\right]$. For $R_i$, it has the lenght at most $q-2$, if $R_i$ only have 2 isolated vertex, we set $w_i$ at $u_i$, otherwise $w_i$ is the center of $R_i$, $2\left(q-i\right)=2\left(q-\frac{q+2}{2}\right)=q-2$, thus $R_i\subseteq N_{q-i}\left[w_i\right]$. By lemma 2.1, we have $b\left(G\right)\le \lceil \sqrt{n}\rceil =q+1$.

When $i\ge \lfloor \frac{q+4}{2}\rfloor$. Since $l\ge q+1$ and $G$ doesn’t have $q$ arms of length $q+1$, then there must exist an arm $C_{l_i,2}$ with the length $\left|l_i\right|\le q$. $C_{l_i,2}-\left\{v_0\right\}$ contain at least 3 vertices, then $N_q\left[x_1\right]$ contains at least $iq+4$ vertices, thus $\left|R\right|\le n-iq-1-3\le q\left(q+2-i\right)-3$. If $i=q+1$, then $\left|R\right|\le q-3\le i=q+1$, the number of vertices is smaller than the number of branches, a contradiction. If $i=q$, then $\left|R\right|\le 2q-3\ge i=q\left(q\ge 3\right)$, a satisfaction.

So we only consider $\lfloor \frac{q+4}{2}\rfloor \le i\le q$, removing a pendant vertex to another pendant vertex doesn’t change the number of vertices of $R$, thus $V\left(R\right)=V\left(P\right)$. By $b\left(R\right)\le b\left(P\right)$ and lemma 2.6, we have

$b\left(R\right)\le b\left(P\right)\le \lfloor \frac{V\left(R\right)}{2i}\rfloor +i\le \lfloor \frac{q\left(q+2-i\right)-3}{2i}\rfloor +i$

Let $f\left(i\right)=\lfloor \frac{q\left(q+2-i\right)-3}{2i}\rfloor +i$, because of the properties of the function, the maximum is attained at the $q$ or $\lfloor \frac{q+4}{2}\rfloor$.

Suppose $i=q$, we have

$b\left(R\right)\le \lfloor \frac{V\left(R\right)}{2i}\rfloor +i=\lfloor \frac{q\left(q+2-i\right)-3}{2i}\rfloor +i=\lfloor \frac{2q-3}{2q}\rfloor +q\le q$

Suppose $i=\lfloor \frac{q+4}{2}\rfloor$, we have

$\begin{array}{c}b\left(R\right)\le \lfloor \frac{V\left(R\right)}{2i}\rfloor +i=\lfloor \frac{q\left(q+2-i\right)-3}{2i}\rfloor +i\\ =\lfloor \frac{q\left(q+2-\lfloor \frac{q+4}{2}\rfloor -3\right)}{2\lfloor \frac{q+4}{2}\rfloor }\rfloor +\lfloor \frac{q+4}{2}\rfloor \end{array}$

when $q$ is even, then $\lfloor \frac{q+4}{2}\rfloor =\frac{q+4}{2}$, we have $b\left(R\right)\le \lfloor \frac{q\left(q+2-\lfloor \frac{q+4}{2}\rfloor -3\right)}{2\lfloor \frac{q+4}{2}\rfloor }\rfloor +\lfloor \frac{q+4}{2}\rfloor =\lfloor \frac{q}{2}-\frac{2q+3}{q+4}\rfloor +\frac{q+4}{2}$, since $\frac{2q+3}{q+4}\in \left(1,2\right)$, then $\lfloor \frac{q}{2}-\frac{2q+3}{q+4}\rfloor =\frac{q}{2}-2$, thus we have $b\left(R\right)\le \frac{q}{2}-2+\frac{q+4}{2}=q$.

When $q$ is odd, then $\lfloor \frac{q+4}{2}\rfloor =\frac{q+4}{2}$, we have $b\left(R\right)\le \lfloor \frac{q\left(q+2-\lfloor \frac{q+4}{2}\rfloor -3\right)}{2\lfloor \frac{q+4}{2}\rfloor }\rfloor +\lfloor \frac{q+4}{2}\rfloor =\lfloor \frac{q}{2}-1\rfloor +\frac{q+3}{2}$, since $\lfloor \frac{q}{2}-1\rfloor =\frac{q-1}{2}-1$, thus we have $b\left(R\right)\le \frac{q-1}{2}-1+\frac{q+3}{2}=q$.

We know $G=N_q\left[x_1\right]\cup R$, suppose $\left(z_1,z_2,\cdots ,z_q\right)$ is a burning sequence of $R$. Next let $x_1=v_0$ and $x_{i+1}=z_i$ for $1\le i\le q$, it is clear that $V\left(G\right)\subseteq N_q\left[x_1\right]\cup N_{q-1}\left[x_2\right]\cup N_{q-2}\left[x_3\right]\cup \cdots \cup N_0\left[x_{q+1}\right]$, by lemma 2.1 we have $b\left(G\right)\le q+1$.

Case 3 If $l\ge 2q$.

When $l\ge 2q$, we proceed with contradiction, assume $G^c$ is the minimal counterexample of octopus graph with order $n$. This means

$b\left(G^c\right)>\lceil \sqrt{\left|b\left(G^c\right)\right|}\rceil =q+1$. Suppose the length of longest arm $C_{l_i,2}$ of $G^c$ is $l$, we have $w_1\in C_{l_i,2}$ such that $d\left(v_0,w_1\right)=l$. Since $l\ge 2q$, we have $w_0\in C_{l_i,2}$ satisfied $d\left(w_0,w_1\right)=q-1$. We set $x_1$ at $w_0$, Clearly, $N_q\left[w_0\right]$ can burn $2q+1$ vertices, We denoted $G_1=G\backslash N_q\left[w_0\right]$ where $v_0\in G_1$, then $\left|G_1|=|G\backslash N_q\left[w_0\right]\right|\le q^2+2q+1-\left(2q+1\right)\le q^2$. Consider $G^c$ is a counterexample of octopus with minimal number of vertices, we have $b\left(G_1\right)\le \lceil \sqrt{\left|G_1\right|}\rceil \le q$, then $\left(z_1,z_2,\cdots ,z_q\right)$ is a burning sequence of $G_1$. Next let $x_1=w_0$, $x_{i+1}=z_i$ for $1\le i\le q$, it is clear that $V\left(G^c\right)\subseteq N_q\left[x_1\right]\cup N_{q-1}\left[x_2\right]\cup N_{q-2}\left[x_3\right]\cup \cdots \cup N_0\left[x_{q+1}\right]$. By lemma 2.1 we have $b\left(G^c\right)\le q+1$. This contradicts to the fact $b\left(G^c\right)>q+1$. Thus, we have $b\left(G\right)\le \lfloor \sqrt{n}\rfloor$. □

Theorem 3.2 If $G=G_{v_0\left(g_1,g_2,a_1\right)}$ is a single tail bicyclic graph with order $n$, then $\lceil \sqrt{n+\frac{21}{4}}-\frac{3}{2}\rceil \le b\left(G\right)\le \lceil \sqrt{n}\rceil$.

Proof. We take a edge $e_i$ from $C_{g_i}$ (i = 1, 2), then we can derive

${G^{\prime }}_{v_0\left(g_1,g_2,a_1\right)}=G_{v_0\left(g_1,g_2,a_1\right)}-\underset{i=1}{\overset{2}{{\displaystyle \sum }}}{e}_i$ is a spider graph. By lemma 2.3 and lemma 2.5, we have $b(G_{v_0\left(g_1,g_2,a_1\right)}\le b\left({G^{\prime }}_{v_0\left(g_1,g_2,a_1\right)}\right)\le \lceil \sqrt{n}\rceil$.

Next, we prove the lower bound, suppose $b\left(G\right)=k$ and $\left(x_1,x_2,\cdots ,x_k\right)$ is an optimal burning sequence of $G$, we set $x_1$ on $v_0$ to contains more vertices, then, $N_{k-1}\left[x_1\right]\le 5\left(k-1\right)+1$, combine with the fact that $\left|N_{k-i}\left[x_i\right]\right|\le 2\left(k-i\right)+1$ for $2\le i\le k$, we get

$\begin{array}{c}\left|N_{k-1}\left[x_1\right]\right|+\underset{i=2}{\overset{k}{{\displaystyle \sum }}}\left(2\left(k-i\right)+1\right)\le \left(5\left(k-1\right)+1\right)+\left(2\left(k-2\right)+1\right)+\cdots +1\\ =\left(5k-4\right)+\left(2k-3\right)+\cdots +1\\ =k^2+3k-3.\end{array}$

by $k^2+3k-3\ge n$, we have $k\ge \lceil \sqrt{n+\frac{21}{4}}-\frac{3}{2}\rceil$. □

Theorem 3.3 If $G=G_{v_0\left(g_1,g_2,a_1,a_2\right)}$ is a double tail bicyclic graph with order $n$, then $\lceil \sqrt{n+8}\rceil -2\le b\left(G\right)\le \lceil \sqrt{n}\rceil$.

Proof. We take a edge $e_i$ from $C_{g_i}$( $i=1,2$), then we can derive ${G^{\prime }}_{v_0\left(g_1,g_2,a_1,a_2\right)}=G_{v_0\left(g_1,g_2,a_1,a_2\right)}-\underset{i=1}{\overset{2}{{\displaystyle \sum }}}{e}_i$ is a spider graph. by lemma 2.3 and lemma 2.5, we have $b\left(G_{v_0\left(g_1,g_2,a_1,a_2\right)}\right)\le b\left({G^{\prime }}_{v_0\left(g_1,g_2,a_1,a_2\right)}\right)\le \lceil \sqrt{n}\rceil$.

Next, we prove the lower bound, suppose $b\left(G\right)=k$ and $\left(x_1,x_2,\cdots ,x_k\right)$ is an optimal burning sequence of $G$. We set $x_1$ on $v_0$ to contains more vertices, then, $N_{k-1}\left[x_1\right]\le 6\left(k-1\right)+1$, combine with the fact that $\left|N_{k-i}\left[x_i\right]\right|\le 2\left(k-i\right)+1$ for $2\le i\le k$, we have

$\begin{array}{c}\left|N_{k-1}\left[x_1\right]\right|+\underset{i=2}{\overset{k}{{\displaystyle \sum }}}\left(2\left(k-i\right)+1\right)\le \left(6\left(k-1\right)+1\right)+\left(2\left(k-2\right)+1\right)+\cdots +1\\ =\left(6k-5\right)+\left(2k-3\right)+\cdots +1\\ =k^2+4k-4.\stackrel{}{}\end{array}$

by $k^2+4k-4\ge n$, we have $k\ge \lceil \sqrt{n+8}\rceil -2$. □

We following discuss the burning number of $G_{v_0\left(g_1,g_2,a_1\right)}$ and $G_{v_0\left(g_1,g_2,a_1,a_2\right)}$.

Consider $G=G_{v_0\left(g_1,g_2,a_1\right)}$, by Theorem 3.2, we have

Corollary 3.4 If $G=G_{v_0\left(g_1,g_2,a_1\right)}$ is a single tail bicyclic graph with order $q^2+p$ for $1\le p\le 2q+1$, then $q-1\le b\left(G\right)\le q+1$.

Next we discuss the graph $G_{v_0\left(g_1,g_2,a_1\right)}$ with burning number $q+1$.

Theorem 3.5 Let $G=G_{v_0\left(g_1,g_2,a_1\right)}$ be a single tail bicyclic graph with order $q^2+p$. If $3q-2\le p\le 2q+1$, then $b\left(G\right)=q+1$

Proof. As we know from the previous, if $b\left(G\right)=q$, then it can contains at most $q^2+3q-2$ vertices of $G$. Since $3q-2\le p\le 2q+1$, then we have $b\left(G\right)\ge q+1$, combine with corollary 3.4, we have $b\left(G\right)=q+1$. □

Theorem 3.6 Let $G=G_{v_0\left(g_1,g_2,a_1\right)}$ be a single tail bicyclic graph with order $q^2+p$ for $1\le p\le 2q+1$. If $g_1\ge q^2$ or $a_1\ge q^2$, then $b\left(G\right)=q+1$.

Proof. We discuss two cases to complete the proof.

Case 1 If $g_1\ge q^2$.

In this case, let $H=C_{g_1+1}$, $\left|H\right|\ge q^2+1$, by lemma 2.2, then $b\left(H\right)\ge q+1$. If $\lceil \frac{g}{2}\rceil \le q$ and $a_1\le q$, then $b\left(G\right)=b\left(H\right)\ge q+1$. If $\lceil \frac{g}{2}\rceil \ge q$ or $a_1\ge q$, then $b\left(G\right)\ge b\left(H\right)\ge q+1$. Thus we have $b\left(G\right)\ge q+1$, combine with corollary 3.4, we have $b\left(G\right)=q+1$.

Case 2 If $a_1\ge q^2$.

In this case, let $H=P_{a_1+1}$, $\left|P_{a_1+1}\right|\ge q^2+1$, by lemma 2.2, $b\left(H\right)\ge q+1$. $H$ is a subgraph of $G$, when $\lceil \frac{g_1}{2}\rceil \le q$, we have $b\left(G\right)=b\left(H\right)\ge q+1$, when $\lceil \frac{g_1}{2}\rceil >q$, we have $b\left(G\right)>b\left(H\right)\ge q+1$. Thus we have $b\left(G\right)\ge q+1$, combine with corollary 3.4, we have $b\left(G\right)=q+1$. □

Theorem 3.7 Let $G=G_{v_0\left(g_1,g_2,a_1\right)}$ be a single tail bicyclic graph with order $q^2+p$ for $1\le p\le 2q+1$. If $\lceil \frac{g_1}{2}\rceil +a_1\ge q^2$ or $\lceil \frac{g_1}{2}\rceil +\lceil \frac{g_2}{2}\rceil \ge q^2$, then

$b\left(G\right)=q+1$. □

Proof. According to the definition of diameter, $d\left(G\right)=\max \left\{\lceil \frac{g_1}{2}\rceil +a_1,\lceil \frac{g_1}{2}\rceil +\lceil \frac{g_2}{2}\rceil \right\}$, it’s clearly that $d\left(G\right)\ge q^2$, by lemma 2.4, we have $b\left(G\right)\ge \lceil \sqrt{d\left(G\right)+1}\rceil \ge \lceil \sqrt{q^2+1}\rceil =q+1$, combine with corollary 3.4, we have $b\left(G\right)=q+1$.

Theorem 3.8 Let $G=G_{v_0\left(g_1,g_2,a_1\right)}$ be a single tail bicyclic graph with order $q^2+p$ for $1\le p\le 2q+1$. If $2q-2\le g_1\le q^2+1$, $2q-2\le g_2\le q^2+1$, $q-1\le a_1\le q^2+1$

1) If $\min \left\{{g^{\prime }}_1,{g^{\prime }}_2,{a^{\prime }}_1\right\}\ne 0$, $\left\{{g^{\prime }}_1,{g^{\prime }}_2,{a^{\prime }}_1\right\}\in J^1\cup J^2\cup J^3\cup J^4\cup J^5$, then $b\left(G\right)=q+1$.

2) If $\min \left\{{g^{\prime }}_1,{g^{\prime }}_2,{a^{\prime }}_1\right\}=0$, $\left\{{g^{\prime }}_1,{g^{\prime }}_2,{a^{\prime }}_1\right\}\in J\left(t\right)$, then $b\left(G\right)=q+1$.

where ${g^{\prime }}_1=g_1-\left(2q-2\right),{g^{\prime }}_2=g_2-\left(2q-2\right),{a^{\prime }}_1=a_1-\left(2q-2\right)$.

Proof. 1) If $\min \left\{{g^{\prime }}_1,{g^{\prime }}_2,{a^{\prime }}_1\right\}\ne 0$, suppose $\left(x_1,x_2,\cdots ,x_q\right)$ is an optimal burning sequence of $G$, we set $x_1$ at $v_0$, then $\left|N_{q-1}\left[x_1\right]\right|=5q-4$, $H=G-N_{q-1}\left[x_1\right]=P_{{g^{\prime }}_1}\cup P_{{g^{\prime }}_2}\cup P_{{a^{\prime }}_1}$. Since $\left\{{g^{\prime }}_1,{g^{\prime }}_2,{a^{\prime }}_1\right\}\in J^1\cup J^2\cup J^3\cup J^4\cup J^5$ , by lemma 2.8, we have $b\left(H\right)=\lceil \sqrt{{g^{\prime }}_1+{g^{\prime }}_2+{a^{\prime }}_1}\rceil +1\le \lceil \sqrt{n-\left(5q-4\right)}\rceil +1\le \lceil \sqrt{{\left(q-1\right)}^2}\rceil +1\le q$, then $b\left(G\right)=q+1$, a contradiction, thus we have $b\left(G\right)\ge q+1$, combine with corollary 3.4, we have $b\left(G\right)=q+1$.

2) If $\min \left\{{g^{\prime }}_1,{g^{\prime }}_2,{a^{\prime }}_1\right\}=0$, suppose $\left(x_1,x_2,\cdots ,x_q\right)$ is an optimal burning sequence of $G$, we set $x_1$ at $v_0$, then $\left|N_{q-1}\left[x_1\right]\right|=5q-4$,

$H=G-N_{q-1}\left[x_1\right]=P_{{g^{\prime }}_1}\cup P_{{g^{\prime }}_2}\cup P_{{a^{\prime }}_1}$. Since $\left\{{g^{\prime }}_1,{g^{\prime }}_2,{a^{\prime }}_1\right\}\in J\left(t\right)$, by lemma 2.7, we have $b\left(H\right)\le \lceil \sqrt{n-\left(5q-4\right)}\rceil +1\le \lceil \sqrt{{\left(q-1\right)}^2}\rceil +1\le q$, then $b\left(G\right)=q+1$, a contradiction, thus we have $b\left(G\right)\ge q+1$. combine with corollary 3.4, we have $b\left(G\right)=q+1$. □

Consider $G=G_{v_0\left(g_1,g_2,a_1,a_2\right)}$, by Theorem 3.3, we have

Corollary 3.9 If $G=G_{v_0\left(g_1,g_2,a_1,a_2\right)}$ is a double tail bicyclic graph with order $q^2+p$ for $1\le p\le 2q+1$, then $q-1\le b\left(G\right)\le q+1$.

Lemma 3.10 If G is disconnected with connected components $G_1,G_2,\cdots ,G_i$, each $G_i$ contains no isolated vertices, then $b\left(G\right)\le b\left(G_1\right)+b\left(G_2\right)+\cdots +b\left(G_i\right)-i+1$.

Proof. For each $G_j\left(1\le j\le i\right)$, we suppose $X_j=\left(x_1^{\left(j\right)},x_2^{\left(j\right)},\cdots ,x_{b\left(G_j\right)}^{\left(j\right)}\right)$ is an optimal burning sequence, Clearly $G$ also has a burning sequence. We claim $X=\left(x_1^{\left(1\right)},\cdots ,x_{b\left(G_1\right)-1}^{\left(1\right)},x_1^{\left(2\right)},\cdots ,x_{b\left(G_2\right)-1}^{\left(2\right)},\cdots ,x_1^{\left(i\right)},\cdots ,x_{b\left(G_i\right)-1}^{\left(i\right)},x_{b\left(G_i\right)}^{\left(i\right)}\right)$ is a burning sequence of $G$.

For each $j\in \left\{1,2,\cdots ,i-1\right\}$, we burn $X\backslash \left\{x_{b\left(G_j\right)}^{\left(j\right)}\right\}$ in order. Before we burn $x_1^{\left(i\right)}$, $C_{i-1}$ need at most 2 rounds to burn complete. Since $G_i$ contains no isolated vertices, then $b\left(G_i\right)\ge 2$. Therefore, when $G_i$ is burned, $G_{i-1}$ has enough time to burn completely, thus all the $G_j$ can be burned completely. since $X$ is a valid burning sequence of $G$, thus we have

$b\left(G\right)\le \left|X\right|=\left(b\left(G_1\right)-1\right)+\cdots +\left(b\left(G_i\right)-1\right)+b\left(G_i\right)=b\left(G_1\right)+b\left(G_2\right)+\cdots +b\left(G_i\right)-i+1$.

□

Next we discuss $G_{v_0\left(g_1,g_2,a_1,a_2\right)}$ with burning number $q+1$.

Theorem 3.11 Let $G=G_{v_0\left(g_1,g_2,a_1,a_2\right)}$ be a double tail bicyclic graph with order $q^2+p$ for $1\le p\le 2q+1$. If $q=2,p=5$, then $b\left(G\right)=q+1$.

Proof. If $q=2$, then $n\in \left[5,9\left],p\in \right[1,5\right]$. If $b\left(G\right)=q$, it can contain at most $q^2+4q-4$ vertices. When $p\ge 4q-3$, then $p=5$, thus $b\left(G\right)\ge q+1$. Combine with corollary 3.9, we have $b\left(G\right)=q+1$. □

Theorem 3.12 Let $G=G_{v_0\left(g_1,g_2,a_1,a_2\right)}$ be a double tail bicyclic graph with order $q^2+p$ for $1\le p\le 2q+1$. If $g_1\ge q^2$ or $a_1+a_2\ge q^2$, then $b\left(G\right)=q+1$.

Proof. We discuss two cases to complete the proof.

Case 1 If $g_1\ge q^2$.

In this case, let $H=C_{g_1+1}$, $\left|H\right|\ge q^2+1$, by lemma 2.2, then $b\left(H\right)\ge q+1$. Because of the symmetry of the circle, we set $x_1$ at $v_0$. If $\max \left\{\frac{g_2}{2},a_1,a_2\right\}\le q$, then $b\left(G\right)=b\left(H\right)\ge q+1$, if $\max \left\{\frac{g_2}{2},a_1,a_2\right\}>q$, then $b\left(G\right)>b\left(H\right)\ge q+1$. Thus we can derive $b\left(G\right)\ge q+1$, combine with corollary 3.9, we have $b\left(G\right)=q+1$

Case 2 If $a_1+a_2\ge q^2$.

In this case, let $H=P_{a_1+a_2+1}$, $\left|P_{a_1+a_2+1}\right|\ge q^2+1$, by lemma 2.2, $b\left(H\right)\ge q+1$. $H$ is a subgraph of $G$, similar to case 1, we have $b\left(G\right)\ge b\left(H\right)\ge q+1$, combine with corollary 3.9, we have $b\left(G\right)=q+1$. □

Theorem 3.13 Let $G=G_{v_0\left(g_1,g_2,a_1,a_2\right)}$ be a double tail bicyclic graph with order $q^2+p$ for $1\le p\le 2q+1$. If $\lceil \frac{g_i}{2}\rceil +a_i\ge q^2$ or $\lceil \frac{g_1}{2}\rceil +\lceil \frac{g_2}{2}\rceil \ge q^2$, then $b\left(G\right)=q+1$.

Proof. According to the definition of diameter,

$d\left(G\right)=\max \left\{\lceil \frac{g_i}{2}\rceil +a_i,\lceil \frac{g_1}{2}\rceil +\lceil \frac{g_2}{2}\rceil \right\}$, it’s clearly that $d\left(G\right)\ge q^2+1$, by lemma 2.4, we have $b\left(G\right)\ge \lceil \sqrt{d\left(G\right)+1}\rceil \ge \lceil \sqrt{q^2+1}\rceil =q+1$, combine with corollary 3.9, we have $b\left(G\right)=q+1$. □

Theorem 3.14 Let $G=G_{v_0\left(g_1,g_2,a_1,a_2\right)}$ be a double tail bicyclic graph with order $q^2+p$ for $1\le p\le 2q+1$. If $2q-2\le g_1\le q^2+1$, $2q-2\le g_2\le q^2+1$, $q-1\le a_1\le q^2+1$, $q-1\le a_2\le q^2+1$

1) If $\min \left\{{g^{\prime }}_1,{g^{\prime }}_2,{a^{\prime }}_1,{a^{\prime }}_2\right\}\ne 0$, $\lceil \sqrt{{g^{\prime }}_1}\rceil \ne 1$, $\lceil \sqrt{{g^{\prime }}_2}\rceil \ne 1$, $\lceil \sqrt{{a^{\prime }}_1}\rceil \ne 1$, $\lceil \sqrt{{a^{\prime }}_2}\rceil \ne 1$, $\lceil \sqrt{{g^{\prime }}_1}\rceil +\lceil \sqrt{{g^{\prime }}_2}\rceil +\lceil \sqrt{{a^{\prime }}_1}\rceil +\lceil \sqrt{{a^{\prime }}_2}\rceil \le q+3$, then $b\left(G\right)=q+1$.