Maximum nullity and zero forcing number on graphs with maximum degree at most three

Abstract

A dynamic coloring of a graph $G$ starts with an initial subset $F\subseteq V\left(G\right)$ of colored vertices, while all the remaining vertices are non-colored. At each time step, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $F$ is called a zero forcing set of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. The zero forcing number of $G$, denoted by $F\left(G\right)$, is the cardinality of a minimum zero forcing set of $G$. The maximum nullity of $G$, denoted by $M\left(G\right)$, is the largest possible nullity over all $\left|V\left(G\right)\right|$ by $\left|V\left(G\right)\right|$ real symmetric matrices $A$ whose non-diagonal entries are non-zero if the corresponding vertices are adjacent in $G$ and with no restriction for its diagonal entries. In this paper, we characterize all graphs $G$ of order $n$, maximum degree at most three, and $F\left(G\right)=3$. Also we classify these graphs with their maximum nullity.

Introduction

Following the notation of [3], [9], for a graph $G$, we define the forcing process as follows: Let $F\subseteq V\left(G\right)$ be an initial set of vertices. At step zero, each vertex in $F$ is colored and any other vertex is non-colored. For $i\ge 1$, at step $i$, each colored vertex $v$ with exactly one non-colored neighbor forces its non-colored neighbor to be colored. At each step $i$, the set of newly get colored vertices is denoted by $V_i$. In particular $V_0=F$. This process will continue until there is no more possible change. At the end of the process, the set of colored vertices is called the derived set. A zero forcing set of $G$ is a set $F\subseteq V\left(G\right)$ of initially colored vertices if the corresponding derived set is equal to $V\left(G\right)$, i.e., $V\left(G\right)=\bigcup V_i$. For each vertex $v\in V\left(G\right)$, the step that $v$ gets colored is denoted by $k_F\left(v\right)$. Note that $k_F\left(v\right)=i$ if and only if $v\in V_i$. The zero forcing number of a graph $G$, denoted by $F\left(G\right)$, is the minimum possible cardinality for a zero forcing set of $G$ and any zero forcing set of $G$ with cardinality $F\left(G\right)$ is named an $F\left(G\right)$-set. Throughout the paper, for brevity, we would rather use “forcing” instead of “zero forcing”. It is simple to see that $F\left(G\right)=\left|V\left(G\right)\right|$ if and only if $G$ is an empty graph i.e., a graph with no edge.

Let $F$ be an $F\left(G\right)$-set. In the forcing process with initial vertex set $F$, any vertex forces at most one vertex while a vertex can be forced by different vertices. Therefore, $V\left(G\right)$ can be partitioned to $\left|F\right|$ chains (sequences) $R_1,\dots ,R_{\left|F\right|}$. For each of these chains, say $R_i=(v_0,v_1,\dots ,v_k)$, $v_0\in F$ and $v_j$ is forced by $v_{j-1}$ for each $j\in \left[k\right]$. The set $\{R_1,\dots ,R_{\left|F\right|}\}$ is called a chain set with respect to $F$ for $G$, which is not necessarily unique. For each $j\in \{0,\dots ,k-1\}$, we define ${next}_{R_i}\left(v_j\right)=v_{j+1}$ and for each $j\in \left[k\right]$, we set ${prev}_{R_i}\left(v_j\right)=v_{j-1}$. As a simple observation, one can see that each chain $R_i=(v_0,v_1,\dots ,v_k)$ induces a path $P_i=v_0{v}_1\dots v_k$ in $G$, i.e, $G\left[V\left(R_i\right)\right]$ is an induced path. To see that, note that if there exist two adjacent vertices $v_{j_1},v_{j_2}\in V\left(R_i\right)$, where $j_2>j_1+1$, then when $v_{j_1}$ forces $v_{j_1+1}$, its neighbor $v_{j_2}$ is non-colored which is impossible. A chain is called trivial if it has only one vertex.

For a graph $G$, a total forcing set is a forcing set of $G$ inducing a subgraph with no isolated vertex. For simplicity, we write TF-set instead of total forcing set. The total forcing number of $G$, denoted by $F_t\left(G\right)$, is the cardinality of a minimum $TF$-set in $G$. Total forcing set was first introduced by Davila [6] as a strengthening of forcing set which was originally introduced by the AIM-minimum rank group [14]. It is observed (for instance, see [6], [8]) that for a graph $G$ with no isolated vertex,

$$F\left(G\right)\le F_t\left(G\right)\le 2F\left(G\right).$$

The forcing process and the forcing number were introduced in [5] and [14] to bound the minimum rank of a graph and hence its maximum nullity. Since then, the forcing number has gained a considerable attention in graph theory and has been related to many graph theoretic parameters. In general, study of forcing number is challenging for many reasons. First, it is difficult to compute, as it is known to be NP-hard [10], [18]. Further, many of the known bounds leave a wide gap for graphs in general. For example, the forcing number of a graph of order $n$ can be as low as $\delta \left(G\right)$ (see [4]), and as high as $n\Delta \left(G\right)∕(\Delta \left(G\right)+1)$ (see [2]).

Gentner and Rautenbach [12] proved that for a graph $G$ with maximum degree $\Delta =\Delta \left(G\right)\le 3$ and girth at least $5$, we have

$$F\left(G\right)\le \left(\frac{n}{2}\right)-\frac{n}{24{log}_{2}n+6}+2,$$

moreover, they introduced two graphs $G_1$ and $G_2$ and proved $F\left(G\right)\le n(\Delta -2)∕(\Delta -1)$ for any connected graph $G⁄\in \{K_{\Delta +1},K_{\Delta ,\Delta },K_{\Delta -1,\Delta },G_1,G_2\}$. For more details and the definitions of $G_1$ and $G_2$, see [12]. They also conjectured that $F\left(G\right)\le \frac{n}{3}+2$ for any graph $G$ with $n$ vertices and $\Delta =3$. In [13], an infinite family of graphs $\left\{G_n\right\}$ with maximum degree $3$ was introduced such that the forcing number of $G_n$ is at least $\frac{4}{9}\left|V\left(G_n\right)\right|$, a counter example to the Gentner–Rautenbach conjecture. Note that, at this point, the best upper bound for the forcing number of connected graphs with maximum degree three [2] is

$$F\left(G\right)\le \frac{n}{2}+1.$$

The equality can be achieved for graphs $K_4$ and $K_{3,3}$. Davila and Henning [7] studied forcing sets and total forcing sets in claw-free cubic graphs. They proved $F\left(G\right)<n∕2$, where $G$ is a connected claw-free cubic graph with $n\ge 10$ vertices. Akbari and Vatandoost [1] gave a partial answer to the problem of determining all graphs $G$ with $M\left(G\right)=F\left(G\right)$ posed by AIM Minimum Rank-Special Graphs Work Group [14], where $M\left(G\right)$ is the maximum nullity of $G$ (for the definition, see Section 1.1). Indeed, they characterized all cubic graphs with forcing number $3$ and, as a corollary, they also showed that $M\left(G\right)=3$ for any graph $G$ in this family.

Problem 1 [14]

Determine all graphs $G$ for which $M\left(G\right)=F\left(G\right)$.

In this paper, we characterize all graphs $G$ with maximum degree at most three and forcing number $3$. We will moreover investigate the maximum nullity of these graphs comparing to their forcing numbers, a partial answer to Problem 1.

For notation and terminology not presented here, we refer readers to [15]. We will use the notation $P_n$ and $K_n$ to respectively denote the path and the complete graph on $n$ vertices. Also we use the standard notation $\left[k\right]=\{1,2,\dots ,k\}$. Let $G$ be a graph. For a path $P=x_0\cdots x_m$ in $G$, we define $x_{i}P=x_i\cdots x_m$ and $P{x}_i=x_0\cdots x_i$ for each $i\in \left[m\right]$.

Let $S_n\left(\mathbb{R}\right)$ be the set of all symmetric $n$ by $n$ matrices over the real numbers. For $A=\left(a_{ij}\right)\in S_n\left(\mathbb{R}\right)$, the graph of $A$, denoted by $\mathcal{G}\left(A\right)$, is a graph with vertex set $\{v_1,\dots ,v_n\}$ and edge set $\left\{v_i{v}_j:a_{ij}\ne 0,1\le i<j\le n\right\}$. It should be noted that the diagonal of $A$ has no role in the definition of $\mathcal{G}\left(A\right)$. The set of symmetric matrices of graph $G$ is $S\left(G\right)=\{A\in S_n\left(\mathbb{R}\right):\mathcal{G}\left(A\right)=G\}$. The minimum rank of a graph $G$, denoted by $mr\left(G\right)$, is the minimum possible rank for a matrix in $S\left(G\right)$ and, similarly, the maximum nullity of $G$, denoted by $M\left(G\right)$, is the maximum nullity of symmetric matrices in $S\left(G\right)$. Clearly, $mr\left(G\right)+M\left(G\right)=n$ and $M\left(G\right)\ge 1$ for any graph $G$.

Johnson et al. [16] defined the graph of $2$- parallel paths. A graph $G$, which is not a path, is said to be a graph of $2$-parallel paths if there are two vertex disjoint induced paths covering all the vertices and $G$ can be drawn in the plane so that these two paths are parallel horizontal lines if we forgot their vertices, and moreover the edges (drawn as segments, not curves) with ends in different paths do not cross. Particularly, union of two disjoint paths is a graph of $2$-parallel paths.

As a generalization, in a natural way, we can define the graph of $k$-parallel paths for any integer $k\ge 1$ as follows: Simply, a $1$-parallel path is just a path. For an integer $k\ge 2$, a graph $G$, which is not a graph of $(k-1)$-parallel paths, is said to be a graph of $k$-parallel paths, if there exist $k$ vertex disjoint induced paths covering all the vertices and $G$ can be drawn in the plane in a way that these paths are parallel horizontal lines if we forgot their vertices, and moreover the edges (drawn as segments, not curves) whose ends are in different paths do not cross each other. Such a drawing is called a standard drawing of $G$, for example see Fig. 1. Note that a graph $G$ may have several standard drawings. Here after, for a fixed standard drawing of a graph of $k$-parallel paths, the $k$ paths used in the definition are called the parallel paths with respect to this drawing and any edge with end-points on different paths is called a segment.

It is clear that any graph of $k$-parallel paths has at least $k$ vertices. Also, It is known that every planar graph can be drawn in the plane such that its edges are segment intersecting only at their end-points, see [11]. Consequently, every planar graph of order $n$ is a graph of $k$-parallel paths for some $k\le n$. Clearly, a non-planar graphs of order $n$ is not a graph of $n$-parallel paths. However, the following assertion indicates that the complete graphs of order $n\ge 6$ are not of $k$-parallel paths for each $k\in \left[n\right]$.

Observation 2

For $n\ge 6$, $K_n$ is not a graph of $k$-parallel paths for each $k\in \left[n\right]$.

Proof

For a contradiction, suppose that $K_n$ is a graph of $k$-parallel paths where $k\in \left[n\right]$. Consider a standard drawing of $K_n$. Since $K_n$ is a complete graph and the parallel paths are induced paths, then each of its parallel paths must be of order one or two. Furthermore, since $K_n$ is not a planar graph, we know $k\le n-1$. Consequently, there must be some parallel path isomorphic to $P_2$. Clearly, there is only one such a $P_2$ path, otherwise, the segments between two $P_2$ paths intersect each others which is not possible. Therefore, there exist $n-2$ parallel paths $P^1,\dots ,P^{n-2}$ isomorphic to $P_1$ and exactly one parallel path $P^{n-1}$ isomorphic to $P_2$. Let $V$ be all the vertices in $K_n$ but a vertex from $P^{n-1}$. In view of the standard drawing of $K_n$, the induced subgraph by $V$ is a planar graph isomorphic to $K_{n-1}$ which is not possible since $n\ge 6$.  ■