Connected zero forcing sets and connected propagation time of graphs

TL;DR

This paper explores the concept of connected zero forcing sets and propagation times in graphs, characterizing graphs with extreme propagation times and analyzing special graph classes and products.

Contribution

It introduces the connected zero forcing set and propagation time, providing characterizations for graphs with extreme propagation times and analyzing specific graph classes.

Findings

Graphs with propagation time |G|-1 and |G|-2 are characterized.

Connected zero forcing sets are studied for special graphs and graph products.

New concepts of connected zero forcing and propagation time are introduced.

Abstract

The zero forcing number $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ with colored (black) vertices which forces the set $V(G)$ to be colored (black) after some times. "color change rule": a white vertex is changed to a black vertex when it is the only white neighbor of a black vertex. In this case, we say that the black vertex forces the white vertex. We investigate here the concept of connected zero forcing set and connected zero forcing number. We discusses this subject for special graphs and some products of graphs. Also we introduce the connected propagation time. Graphs with extreme minimum connected propagation times and maximum propagation times $|G|-1$ and $|G|-2$ are characterized.

Equations

$$Z_c(G⟨H_1,H_2,\dots ,H_n⟩)\le |G|+\sum _{i=1}^{n}Z(H_i).$$

$$Z_c(G⟨H_1,H_2,\dots ,H_n⟩)=|G|+\sum _{i=1}^{n}Z(H_i).$$

$$Z_c(G⟨H_1,H_2,\dots ,H_n⟩)\le |G|+\sum _{i=1}^{n}Z(H_i).$$

Full text

MSC 2010: 05C50, 05C12

KEYWORDS: zero forcing number; connected zero forcing number; propagation time

1 Introduction

Let $V(G)$ and $E(G)$ be the vertex set and the edge set of a graph $G=(V,E)$, respectively. For a vertex $v\in V(G)$, the $open\ neighborhood$ $N(v)$ is the set $\{u\in V(G):uv\in E(G)\}$. The $closed\ neighborhood$ $N[v]=N(v)\cup\{v\}$. Simple graph containing no graph loops or multiple edges. In this paper each graph is undirected, finite, simple and the vertex set is nonempty. A graph $G_{0}=(V_{0},E_{0})$ is a $subgraph$ of graph $G=(V,E)$ if $V^{\prime}\subseteq V$ and $E^{\prime}\subseteq E$.

A $complete\ graph$ is a graph in which every two distinct vertices are adjacent. The complete graph on $n$ vertices is denoted by $K_{n}$. A graph $(V,E)$ is $bipartite$ if the vertex set $V$ can be partitioned into two nonempty subsets $X$ and $Y$ such that every edge of graph has one vertex (endpoint) in $X$ and one in $Y$. A $complete\ bipartite\ graph$ is bipartite graph $K_{n,m}$ with $|X|=n,\ |Y|=m$ and $E=\{\{u,w\}:u\in U,w\in W\}$. The $path$ is the graph $P_{n}$ with $V(P_{n})=\{v_{1},v_{2},...,v_{n}\}$ and $E(P_{n})=\{v_{1}v_{2},v_{2}v_{3},...,v_{n-1}v_{n}\}$. The $cycle$ is the graph $C_{n}$ with $V(C_{n})=\{v_{1},v_{2},...,v_{n}\}$ and $E(C_{n})=\{v_{1}v_{2},v_{2}v_{3},...,v_{n-1}v_{n},v_{n}v_{1}\}$.

The following graph operations are used to construct families of graphs:

$\bullet$ The $Cartesian\ product$ of two graphs $G$ and $H,$ is denoted by $G\square H,$ is the graph with vertex set $V(G)\times V(H)$ such that $(u,v)$ is adjacent to $(u^{\prime},v^{\prime})$ if and only if $(1)u=u^{\prime}$ and $vv^{\prime}\in E(H),$ or $(2)v=v^{\prime}$ and $uu^{\prime}\in E(G).$

$\bullet$ The $strong\ product$ of two graphs $G$ and $H,$ is denoted by $G\boxtimes H,$ is the graph with vertex set $V(G)\times V(H)$ such that $(u,v)$ is adjacent to $(u^{\prime},v^{\prime})$ if and only if $(1)u=u^{\prime}$ and $vv^{\prime}\in E(H),$ or $(2)v=v^{\prime}$ and $uu^{\prime}\in E(G),$ or $(3)uu^{\prime}\in E(G)$ and $vv^{\prime}\in E(H).$

$\bullet$ The $corona\ product$ of two graphs $G$ and $H,$ denoted $G\circ H,$ is the graph of order $|G||H|+|G|$ obtained by taking one copy of $G$ and $|G|$ copies of $H,$ and jointing all the vertices in the $i$th copy of $H$ to the $i$th vertex of $G.$

$\bullet$ The $generalized\ corona$ of $G$ with $H_{1},H_{2},\ldots,H_{n},$ is defined as the graph obtained by joining all vertices of $H_{i}$ to the $i-$th vertex of $G.$ We denote this graph by $G\langle H_{1},H_{2},\ldots,H_{n}\rangle.$

$\bullet$ The $n$th $supertraingle$ is denoted by $T_{n}$ is an equilateral triangular grid such that each side of it contains $n$ vertices (see Figure 1.3).

$\bullet$ The $wheel\ graph$ $W_{n}$ of order $n$ ($n$-wheel) is a graph that contains a cycle of order $n-1$, and for which every graph vertex in the cycle is connected to one other graph vertex.

$\bullet$ The $star\ graph$ $S_{n}$ of order $n$ ($n$-star) complete bipartite graph $K_{1,n-1}$.

Zero forcing sets and the zero forcing number were introduced in [1]. The zero forcing number is a useful tool for determining the minimum rank of structured families of graphs and small graphs, and is motivated by simple observations about null vectors of matrices.

Definition 1.1

[1]** Color-change rule: Let $G$ be a graph with each vertex colored either white or black, $u$ be a black vertex of $G,$ and exactly one neighbor $v$ of $u$ be white. Then change the color of $v$ to black. When this rule is applied, we say $u$ forces $v,$ and write $u\rightarrow v.$

Definition 1.2

[1]** A $zero\ forcing\ set$ ( or $ZFS$ for brevity) of a graph $G$ is a subset $Z$ of vertices such that if initially the vertices in $Z$ are colored black and remaining vertices are colored white, the entire graph $G$ may be colored black by repeatedly applying the color-change rule. The zero forcing number of $G,Z(G),$ is the minimum size of a zero forcing set. Any zero forcing set of order $Z(G)$ is called a minimum zero forcing set.

For a coloring of $G$, the $derived\ coloring$ is the result of applying the color-change rule until no more changes are possible. For the black set of vertices $B$, the derived coloring is denoted by $der(B)$ and it is unique [1].

Example 1.3

In following figure, we see that $Z(T_{4})=4$ and $Z(C_{8})=2$.

Definition 1.4

[11]** Let $B$ be a zero forcing set of $G$ and $B^{0}=B$. For $t\geq 0$, we define $B^{(t+1)}$ to be the set of vertices $w$ for which there exists a vertex $b\in\cup_{s=0}^{t}{B^{(s)}}$ such that $w$ is the only neighbor of $b$ not in $\cup_{s=0}^{t}{B^{(s)}}$. $The\ propagation\ time$ of $B$ in $G$, denoted $pt(G,B)$, is the smallest integer $t^{\prime}$ such that $V=\cup_{s=0}^{t^{\prime}}{B^{(s)}}$.

In the other word, the propagation time is the number of steps it take for an initial zero forcing set to force all vertices of a graph to black.

It is possible that two minimum zero forcing set of one graph have the different propagation time (see Example 1.2 of [11]). Zero forcing parameters were studied and applied to the minimum rank problem and quantum systems [6, 7, 11]. Also, this concept was named as graph infection or graph propagation [1]. For more information about the propagation time, see [2, 3, 4, 6, 7, 9, 10, 12, 14]. This parameter is investigated for graphs with extra condition or some type of the zero forcing set [8, 15].

Definition 1.5

[11]** The minimum propagation time of $G$ is $pt(G)=min\{pt(G,B)|B$ is a minimum zero forcing set of $G\}$.

Definition 1.6

[11]** The maximum propagation time of G is defined as $PT(G)=max\{pt(G,B)|B$ is a minimum zero forcing set of $G\}.$

Note that $Z(G)$ and $pt(G)$ are not subgraph monotone. That is, a graph may have a subgraph with greater zero forcing number or minimum propagation time, as we see in the following example. Also, see the Example 1.5 of [11].

Example 1.7

In Figure 1.3, Let $ZFS(T_{4})=\{a,g,h,c\}$, $ZFS(C_{8})=\{a,g\}$ and $ZFS(P_{6})=\{a\}$ Let $B_{0}=ZFS(T_{4})=\{a,g,h,c\}$, $B_{1}=ZFC(C_{8})=\{a,g\}$ and $B_{2}=ZFC(P_{6})=\{a\}$. Then we have:

[TABLE]

So, $pt(T_{4},B_{0})=4$, $pt(C_{8},B_{1})=3$ and $pt(P_{6},B_{2})=5$. It is easy to see that The minimum propagation time of $T_{4}$ is $4$, the minimum propagation time of its subgraph $C_{8}$ is $3$ and the minimum propagation time of its subgraph $P_{6}$ is $5$. So, this concept is not subgraph monotone (The cycle $C_{8}$ and the path $P_{6}$ are subgraphs of the graph $T_{4}$.).

2 Connected zero forcing set

In this section, we investigate the concept of connected zero forcing. We do a comparison between the zero forcing set and connected zero forcing set for special graphs. In [5], it is mentioned that $R\subseteq V(G)$ is a connected forcing set of G, If the subset $R$ is forcing set and it induces a connected subgraph. Now we have the following definition.

Definition 2.1

A $connected\ zero\ forcing\ set$ ( or $CZFS$ for brevity) of a graph $G,$ is a zero forcing set such that be connected in components of $G$. The $connected\ zero\ forcing\ number$ of $G,Z_{c}(G),$ is the minimum size of connected zero forcing sets. Any connected zero forcing set of order $Z_{c}(G)$ is called a minimum connected zero forcing set.

Applying Definition , we obtain the following proposition.

Proposition 2.2

Every CZFS is a ZFS and if a ZFS is connected in components, then it is a CZFS.

Corollary 2.3

For any graph $G,\,Z(G)\leq Z_{c}(G).$

It may be interesting to compare the zero forcing number and connected zero forcing number of some well known graphs:

Example 2.4

In Figure 1, we see the ZFS and CZFS for the star graph $S_{6}$.

From definition, we can easily observe that these zero forcing sets and connected zero forcing sets are minimum. Thus, we have the following proposition for the star graphs and other well known graphs.

Proposition 2.5

$Z(K_{n})=Z_{c}(K_{n})=n-1;$* $Z(P_{n})=Z_{c}(P_{n})=1;$ $Z(C_{n})=Z_{c}(C_{n})=2;$

$Z(W_{n})=Z_{c}(W_{n})=3;$ $Z(T_{n})=Z_{c}(T_{n})=n;$ $Z(S_{n})=n-2,\,Z_{c}(S_{n})=n-1.$*

Theorem 2.6

$Z(K_{n_{1},n_{2},\cdots,n_{t}})=Z_{c}(K_{n_{1},n_{2},\cdots,n_{t}})=n_{1}+n_{2}+\cdots+n_{t}-2.$**

**proof. **Denote the parts of graph by $U_{1}$, $U_{2}\dots,U_{t}$. Let $v_{1}\in U_{1}$. If this vertex is black and has $n_{2}+n_{3}+\dots+n_{t}-1$ black neighbors, then it can change the color of another vertex. But the color of vertices $U_{1}-\{v_{1}\}$ can be changed by a vertex of another part for example $v_{2}\in U_{2}$. For this aim, all neighbors of $v_{2}$ must be black, except one of them. Therefore $U_{1}-\{v_{1}\}$ at most has one white vertex that is, $Z(K_{n_{1},n_{2},\cdots,n_{t}})=Z_{c}(K_{n_{1},n_{2},\cdots,n_{t}})=n_{1}+n_{2}+\cdots+n_{t}-2.$

Now, we find a bound for connected zero forcing number of different product of two graphs. First the strong product of a cycle and a path, we give a simple example. The general case can be found by a similar way.

Example 2.7

In the following figure, we see a CZFS of size $n+2m-2=5+6-2=9$, for $C_{n}\boxtimes P_{m},n=5$ and $m=3$.

Theorem 2.8

$Z(C_{n}\boxtimes P_{m})$* and $Z_{c}(C_{n}\boxtimes P_{m})\leq n+2m-2.$*

Theorem 2.9

• •

$Z_{c}(G\square P_{t})=Z(G\square P_{t})=|G|.$**

• •

$Z_{c}(G\square H)\leq min\{Z_{c}(G)|H|,Z_{c}(H)|G|\}$.

**proof. **The proof is resulted from definitions.

Theorem 2.10

$Z(P_{n}\boxtimes P_{m})=Z_{c}(P_{n}\boxtimes P_{m})\leq n+m-1.$**

**proof. **The graph $P_{n}\boxtimes P_{m}$ is a $n\times m$ grid. We know that $n+m-1\geq Z(P_{n}\boxtimes P_{m})$ [1]. Now consider the CZFS contains all vertices of the first column and the first row of this grid. This set is connected and ZFS. The size of this set is $n+m-1$.

Theorem 2.11

Let $G$ be a connected graph with $|G|=n.$ Then

[TABLE]

In particular, if $H_{i}$’s are connected and $|H_{i}|>1$ for all $i,$ then

[TABLE]

**proof. **Let $Z_{i}$ be a minimum zero forcing set of $H_{i}$ for all $1\leq i\leq n$. It is easy, to see that $V(G)\cup Z_{1}\cup\ldots\cup Z_{n}$ is a ZFS for $G\langle H_{1},H_{2},\ldots,H_{n}\rangle$. Also this set is connected, because the graph $G$ is connected. Thus

[TABLE]

Now, we claim that if $|H_{i}|>1$ for all $i,$ then this set is minimum. First notice that every CZFS contains all vertices of $G.$ Because if a vertex of G such as $v_{i}$ is initially white, then by connectedness of zero forcing set, all of the vertices of $H_{i}$ must be white or the CZFS is a subset of $V(G)$. In the first case, forcing of vertices of $H_{i}$ should be begin with a force by $v_{i}$, which is not possible where $|H_{i}|>1$. In the second case, there exists an vertex $v_{i}$ such that it is the white neighbor of each vertex in $H_{i}$. All vertices in $H_{i}$ should be black and the rest of graph is a path. Thus, it easily follows that $|G|=1$ and $H_{i}$ is complete graph. So the result follows.

Now, if $v_{i}\rightarrow u$ is a force on this graph, it means that $u$ is the only white vertex in $H_{i}$. The graph $H_{i}$ is connected, so there exists another vertex $u_{0}\in H_{i}$ which is a neighbor of $u$. Thus we can replace $v_{i}\rightarrow u$ by $u_{0}\rightarrow u$. Therefore, all the forces are in $H_{i}$. So, for every minimum CZFS $A$ of $G\langle H_{1},H_{2},\ldots,H_{n}\rangle$, $A\cap V(H_{i})$ is a ZFS for $H_{i}$. Now, it follows that $V(G)\cup Z_{1}\cup\cdots\cup Z_{n}$ is minimum.

Corollary 2.12

For any graph $G$ and $H,$ we have

• •

$Z(G\circ H)\leq Z(G)+|G|Z(H).$**

• •

$Z_{c}(G\circ H)\leq Z_{c}(G)+|G|Z(H).$**

Remark 2.13

The integer of $Z_{c}(G)-Z(G)$ can be very large, for example if $G=C_{n}$ and $H=P_{m}$ where $m>1,$ then $Z(G\circ H)=n+2$ and $Z_{c}(G\circ H)=2n$ while if we set $G=P_{n}$ and $H=C_{m},\ m>2$ then $Z(G\circ H)=2n+1$ and $Z_{c}(G\circ H)=3n$.

3 Connected propagation time

In this section, we introduce the concept of connected propagation time for graph $G$. We characterize graphs $G$ having extreme minimum and maximum connected propagation time $|G|-1$, $|G|-2$ and [math].

Definition 3.1

Let $B$ be a connected zero forcing set of $G$ and $B^{0}=B$. for $t\geq 0$, $B^{(t+1)}$ is the set of vertices $w$ for which there exists a vertex $b\in\cup_{s=0}^{t}{B^{(s)}}$ such that $w$ is the only neighbor of $b$ not in $\cup_{s=0}^{t}{B^{(s)}}$. The connected propagation time of $B$ in $G$, denoted by $pt_{c}(G,B)$, is the smallest integer $t^{\prime}$ such that $V=\cup_{s=0}^{t^{\prime}}{B^{(s)}}$.

Definition 3.2

The minimum connected propagation time of $G$ is $pt_{c}(G)=min\{pt_{c}(G,B)|B$ is a minimum connected zero forcing set of $G\}$.

Definition 3.3

The maximum connected propagation time of G is defined as $PT_{c}(G)=max\{pt_{c}(G,B)|B$ is a minimum connected zero forcing set of $G\}.$

Here, the case of connected propagation time $|G|-1$ and $|G|-2$ are investigated.

Theorem 3.4

Let $G$ be a connected graph. Then the following conditions are equivalent:

• •

$Z_{c}(G)=1;$**

• •

$pt_{c}(G)=|G|-1;$**

• •

$Pt_{c}(G)=|G|-1;$**

• •

$G$* is a path.*

**proof. **We prove the sequence $(ii)\rightarrow(iii)\rightarrow(i)\rightarrow(iv)\rightarrow(ii).$

First, note that for each graph $G,$ we have $pt_{c}(G)\leq Pt_{c}(G)\leq|G|-Z_{c}(G)\leq|G|-1.$ Thus, from $(ii),$ we can deduce $(iii),$ and if $(iii)$ holds, then $Z_{c}(G)=1.$ In addition, if $Z_{c}(G)=1,$ then $Z(G)=1$ and therefore $G$ is a path. The last part is trivial.

Definition 3.5

For $k\geq 1$ and nonnegative integer numbers $n_{1},\cdots,n_{k}$ ($n_{i}$s can be zero), by $PC(n_{1},n_{2},\cdots,n_{k})$ we mean a collection of graphs which contains a path $P_{k+2}=(v_{1},\cdots,v_{k+2})$ and $k$ cycles, with the following conditions:

• •

the $i$th cycle ($1\leq i\leq k$) is $(v_{i},v_{i+1},v_{i+2},u^{i}_{1},\dots,u^{i}_{n_{i}})$ with $n_{i}$ new vertices, which are probably neighbors of $v_{i+1}$.

• •

For each $i,j,s,t$ we have $u^{i}_{j}\neq u^{t}_{s}$ and $u^{i}_{j}\neq v_{t}$.

For instance, the following figures are PC(3) and PC(4,3,5,0,2).

Definition 3.6

Let $G$ and $H$ be disjoint graphs, each with a vertex labeled $v$. Then $G\oplus_{v}H$ is the graph obtained by identifying the vertex $v$ in $G$ with the vertex $v$ in $H$.

Theorem 3.7

Let $G$ be a graph with $Pt_{c}(G)=|G|-2$. Then

If $G$ is not connected, then $G=K_{1}\dot{\cup}P_{|G|-1}$. 2. 2.

If $G$ is connected, then $G$ is of the form $PC(n_{1},n_{2},\cdots,n_{k-1},0)$ or $PC(n_{1},n_{2},\cdots,n_{k})\oplus_{v_{k+1}}P_{m}$ where $(v_{1},\dots,v_{k+2})$ is the path of PC and $P_{m}$ is a path with $m\geq 2$ and pendent vertex $v_{k+1}$.

**proof. **From $Pt_{c}(G)\leq|G|-Z_{c}(G),$ it follows that $Z_{c}(G)\leq 2.$ If $Z_{c}(G)=1$ then by previous theorem $Pt_{c}(G)=|G|-1,$ which is a contradiction. So, $Z_{c}(G)=2.$

If $G$ is not connected then it contain two component, each of them has the CZFS of size one thus each component is one path and in every stage, only the color of one vertex will be changed. So, one path is $P_{1}=K_{1}$ and the graph $G$ is $G=K_{1}\dot{\cup}P_{|G|-1}$.

If $G$ is connected, then the connected zero forcing set is consist of two vertices $v_{1}$ and $v_{2}$ which are adjacent by an edge $e$. Observe that in each stage, only the color of one vertex will be changed. The edge $e$ is not a cutting- edge, otherwise $G\setminus\{e\}$ has two components $G_{1}$ and $G_{2}$ with $v_{1}\in V(G_{1})$ and $v_{2}\in V(G_{2}),$ thus $\{v_{1}\}$ is a zero forcing set for $G_{1}$ and $\{v_{2}\}$ is a zero forcing set for $G_{2}.$ Therefore, $G_{1}$ and $G_{2}$ are path and so $G$ is a path and $\{v_{1},v_{2}\}$ is not an minimum connected zero forcing set. Thus, there is another path between $v_{1}$ and $v_{2}$.

Consider a path between $v_{1}$ and $v_{2}$ with largest possible length, namely, $P:v_{1},u_{1},u_{2},\cdots,u_{n},v_{2}$. Just one vertex of $v_{1}$ or $v_{2}$ can have neighborers in $V(G)-V(P)$. Otherwise $v_{1}$ and $v_{2}$ can not change the color of vertices $V(G)-\{v_{1},v_{2}\}$. Because both of them have at least two white neighbors.

Without loss of generality, let $v_{1}$ has no neighbors except $u_{1}$ and $v_{2}$. There exist two cases for $v_{2}$.

1. Case 1.

If $v_{2}$ has another neighbors which does not belong to this path, then in the first step $v_{1}$ forces $u_{1}$. Again, $u_{1}$ can not have more than one white neighbor, which is $u_{2}$. So, the only neighbors of $u_{1}$ are $v_{1},u_{2}$ and probably $v_{2}$. Continuing this way, in the $i$th step, $u_{i-1}$ forces $u_{i}$ and all of the $u_{i}$s with $i<n$ are adjacent to $u_{i-1},u_{i+1}$ and probably $v_{2}$. After $n$ steps, the set $\{u_{n},v_{2}\}$ is a zero forcing set for $G\setminus\{v_{1},u_{1},\cdots,u_{n-1}\}$ and since the path $P$ has maximum length, it follows that there is no other path except the edge $\{u_{n},v_{2}\}$ between these two vertices $G\setminus\{v_{1},u_{1},\cdots,u_{n-1}\}$. This means that this edge, is a cutting edge. Since in each step only one vertices can be forced by black vertices, it can be deduced that $u_{n}$ has no other neighbors and a path can be jointed to $v_{2}$. Putting $v_{3}=u_{n}$, $G$ is of the form $PC(n-1)\oplus_{v_{2}}P_{m}$ with $m\geq 2$ (see following figure).

\Rightarrow$$PC(n-1)\oplus_{v_{2}}P_{m}$$u_{1}$$v_{2}$$v_{1}$$u_{n}$$v_{1}$$v_{3}=u_{n}$$v_{2}$$u_{n-1}$$u_{1} 2. Case 2.

All of the neighbors of $v_{2}$ belongs to the path. If $n=1$, put $v_{3}=u_{1}$ and $G=PC(0)=K_{3}$, else let $1\leq n_{1}<n$, be the largest number for which $u_{n_{1}}$ is adjacent to $v_{2}$. Put $v_{3}=u_{n_{1}}$. As previous part, it can be seen that each $u_{i}\ (1\leq i\leq i_{2})$ are at most of degree 3 with two neighbors of the path and probably $v_{2}$ and in the $i$th step, $u_{i-1}$ forces $u_{i}$.

Now $\{v_{2},v_{3}\}$ is a CZFS for $G\setminus\{v_{1},u_{1},\cdots,u_{{n_{1}}-1}\}$ such that in each step only one vertex becomes black. So by repeating the proof, there exist two cases, either $G$ is of the form $PC(n_{1}-1,n-n_{1})\oplus_{v_{3}}P_{m}$ with $v_{4}=u_{n_{1}+1}$ or all neighbors of $v_{3}$ belongs to the path. In the first case the graph has form $A$ (see following figure).

Av_{1}$$v_{4}$$v_{2}$$v_{3}

When all neighbors of $v_{3}$ belong to the path, if $n-n_{1}=1$, then put $v_{4}=u_{n_{1}+1}$ and $G=PC(n_{1}-1,0)$ and graph is of the form $B$ (see following figure). Else we can choose $v_{4}=u_{n_{2}}$, such that $n_{1}+1<n_{2}\leq n$ and $n_{2}$ is the smallest number for which $u_{n_{2}}$ is adjacent to $v_{3}$.

$\Rightarrow$Bu_{1}$$v_{2}$$v_{1}$$u_{n}$$u_{n_{1}}$$v_{1}$$v_{4}=u_{n_{1}}+1$$v_{2}$$v_{3}

Going on, we obtain a path $v_{1},v_{2},\cdots,v_{k+2}$ with $v_{i}=u_{n_{i-2}}$ and $1\leq n_{1}<n_{3}<\dots<n_{4}<n_{2}<n$ and $G$ is of the form $PC(n_{1}-1,n-n_{2},n_{3}-n_{1}-1,n_{2}-n_{4}-1,...,n_{k-1}-n_{k-3}-1,|n_{k}-n_{k-2}|-1)\oplus_{v_{k+1}}P_{m}$ or $|n_{k}-n_{k-2}|=1$. Notice that $n_{1}-1$ is the number of vertices between $v_{1}$ and $v_{3}$. $n-n_{2}=n-(n_{1}+1)$ is the number of vertices between $v_{2}$ and $v_{4}$, in the fact $i$th integer is the number between $v_{i}$ and $v_{i+2}$. So, $G$ is of the form $PC(n_{1}-1,n-n_{2},n_{3}-n_{1}-1,n_{2}-n_{4}-1,...,n_{k-1}-n_{k-3}-1,|n_{k}-n_{k-2}|-1,0)$.

Theorem 3.8

For connected graph $G,$ we have $pt_{c}(G)=|G|-2$ if and only if $G$ is as Theorem 3.7 and in addition

If $G=PC(n)\oplus_{v_{2}}P_{m}$ for $m\geq 1$, with cycle $v_{1},u_{1},\cdots,u_{n},v_{3},v_{2}$, then $u_{1}$ and $u_{n}$ are adjacent to $v_{2}$. 2. 2.

If $G=PC(n_{1},n_{2},\cdots,n_{k})\oplus_{v_{k+1}}P_{m}$ for some $k>1$ and $m\geq 1$, then in the first cycle $v_{1},u_{1},\cdots,u_{n},v_{3},v_{2}$, we have $u_{1}$ is adjacent to $v_{2}$.

**proof. **If $pt_{c}(G)=|G|-2,$ then $Pt_{c}(G)\geq|G|-2.$ From Theorem 3.4, it follows that $Pt_{c}(G)=|G|-2.$ Thus, $G$ needs to be of one of the forms described in Theorem 3.7.

One should note that if $u_{1}$ in cycle $v_{1},u_{1},\cdots,u_{n_{1}},v_{3},v_{2}$, (first cycle) not to be a neighbor of $v_{2}$ then the set $\{v_{1},u_{1}\}$ is a CZFS of $G$ and in the first step two vertices $u_{2}$ and $v_{2}$ become black. So, $pt_{c}(G)<|G|-2$.

Similarly, if $k=1$, the set $\{v_{3},u_{n}\}$ is another CZFS of $G$. So, the vertex $u_{n}$ must be adjacent to $v_{2}$.

Conversely, if $G$ has the above conditions and $k>1$, we can easily seen that even all the vertices of $i$th cycle with $1<i$ can not be a ZFS. Except in the case that $m=1$, in the $k$th cycle each of the $\{v_{k+1},v_{k+2}\}$ or $\{v_{k},v_{k+2}\}$ is a CZFS with propagation time $|G|-2$. In the case that $k=1$, if $m=1$, then $G=K_{3}$.

Let $m>1$ and $u_{1}$ and $u_{n}$ are adjacent to $v_{2}$. Then for each $1\leq i\leq n-1$, the set $\{u_{i},u_{i+1}\}$ can not be a ZFS. To see this, let $1\leq j_{1}\leq i<i+1\leq j_{2}\leq n$ such that $j_{1}$ is the largest number not greater than $i$ such that $u_{j_{1}}$ is adjacent to $v_{2}$ and $j_{2}$ is the smallest number greater than $i$ for which $u_{j_{2}}$ is adjacent to $v_{2}$. Then it is easily seen that $\{u_{j_{1}},u_{j_{1}+1},\cdots,u_{j_{2}}\}$ can not be a ZFS (see following figure, case $A$).

In addition, for each $i$, if $u_{i}$ is a neighbor of $v_{2}$, then the set $\{u_{i},v_{2}\}$ is not a ZFS (see following figure, case $B$). Because both of them have degree greater than two. Therefore, the only minimum CZFS’s are $\{v_{1},v_{2}\},\ \{u_{1},v_{1}\},\ \{v_{3},v_{2}\},$ and $\{u_{n},v_{3}\}$ with propagation times $|G|-2$.

A$$B$$v_{1}$$v_{3}$$u_{j_{2}}$$v_{2}$$u_{i+1}$$u_{i}$$v_{1}$$v_{3}$$u_{i}$$v_{2}

Remark 3.9

In [15], Row investigated the graph with $Z(G)=2$. He proved that $Z(G)=2$ if and only if $G$ is a graph of two parallel paths. In his proof, he used a result of [3] for characterization the graphs with maximum nullity 2.

Applying Row’s theorem, in [11], the authors define a special type of graphs with two parallel paths, name zigzag graphs, and show that a connected graph with $pt(G)=|G|-2$ is a zigzag graph with special conditions.

Of course the graphs are introduced in Theorem 3.7 and Corollary 3.8, are a special type of the zigzag graph and since every graph with $Z_{c}(G)=2$ satisfies $Z(G)=2$, we can state a proof based on these known results. But here we present an elementary and straightforward proof for these theorems.