There are profound relations between the zero forcing number and minimum rank of a graph. We study the relation of both parameters with a third one, the algebraic co-rank; that is defined as the largest $i$ such that the $i$-th critical ideal is trivial. This gives a new perspective for bounding and computing these three graph parameters.
The minimum forcing number of a graph $G$ is the smallest number of edges simultaneously contained in a unique perfect matching of $G$. Zhang, Ye and Shiu \cite{HDW} showed that the minimum forcing number of any fullerene graph was bounded below by $3$. However, we find that there exists exactly one excepted fullerene $F_{24}$ with the minimum forcing number $2$. In this paper, we characterize all fullerenes with the minimum forcing number $3$ by a construction approach. This also solves an open problem proposed by Zhang et al. We also find that except for $F_{24}$, all fullerenes with anti-fo…
Let $M$ be a perfect matching in a graph. A subset $S$ of $M$ is said to be a forcing set of $M$, if $M$ is the only perfect matching in the graph that contains $S$. The minimum size of a forcing set of $M$ is called the forcing number of $M$. Pachter and Kim [Discrete Math. 190 (1998) 287--294] conjectured that the forcing number of every perfect matching in the $n$-dimensional hypercube is at least $2^{n-2}$, for all $n \ge 2$. Riddle [Discrete Math. 245 (2002) 283-292] proved this for even $n$. We show that the conjecture holds for all $n \ge 2$. The proof is based on simple linear algebra.
Let $G$ be a simple, finite, and undirected graph with vertices each given an initial coloring of either blue or white. Zero forcing on graph $G$ is an iterative process of forcing its white vertices to become blue after a finite application of a specified color-change rule. We say that an initial set $S$ of blue vertices of $G$ is a zero forcing set for $G$ under the specified color-change rule if a finite number of iterations of zero forcing results to an updated coloring where all vertices of $G$ are blue. Otherwise, we say that $S$ is a failed zero forcing set for $G$ under the specified c…
Leaky-forcing is a recently introduced variant of zero-forcing that has been studied for families of graphs including paths, cycles, wheels, grids, and trees. In this paper, we extend previous results on the leaky forcing number of the d-dimensional hypercube, $Q_d$, to show that the $(d-2)$-leaky forcing number of $Q_d$ is at most $2^{d-1}$. We also examine a question about the relationship between the size of a minimum $\ell$-leaky-forcing set and a minimum zero-forcing set for a graph $G$.
Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks. In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated…
Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Among all perfect matchings $M$ of $G$, the minimum and maximum values of $f(G,M)$ are called the minimum and maximum forcing numbers of $G$, denoted by $f(G)$ and $F(G)$, respectively. Then $f(G)\leq F(G)\leq n-1$. Che and Chen (2011) proposed an open problem: how to characterize the graphs $G$ with $f(G)=n-1$. Later they showed that for a bipartite graph $G$, $f(G…
It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of certain circulant graphs, including some bipartite circulants, cubic circulants, and circulants which are torus products, to obtain bounds on the minimum rank and the maximum nullity. We also evaluate when the zero forcing number will give equality.
The minimum rank of a simple graph $G$ is the smallest possible rank over all symmetric real matrices $A$ whose nonzero off-diagonal entries correspond to the edges of $G$. Using the zero forcing number, we prove that the minimum rank of the butterfly network is $\frac19\left[(3r+1)2^{r+1}-2(-1)^r\right]$ and that this is equal to the rank of its adjacency matrix.
Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique perfect matching is $n^2$. We know that $G$ has a unique perfect matching if and only if $f(G)=0$. Along this line, we generalize the classical result to all graphs $G$ with $f(G)=k$ for $0\leq k\leq n-1$, and characterize corresponding extremal graphs as well. Hence we get a non-trivial lower bound of $f(G)$ in terms of the order and size. For bipartite graph…
The forcing number of a perfect matching of a graph was introduced by Harary et al., which originated from Klein and Randić's ideal of innate degree of freedom of Kekulé structure in molecular graph. On the opposite side in some sense, Vukičević and Trinajstié proposed the anti-forcing number of a graph, afterwards Lei et al. generalized this idea to single perfect matching. Recently the forcing and anti-forcing polynomials of perfect matchings of a graph were proposed as counting polynomials for perfect matchings with the same forcing number and anti-forcing number respectively. In this paper…
Zero forcing is a one-player game played on a graph. The player chooses some set of vertices to color, then iteratively applies a color change rule: If all but one of a colored vertex's neighbors are colored, color (i.e. "force") the remaining uncolored neighbor. Generally, the goal is to find the minimum number of vertices to initially color such that all vertices eventually become colored. Recently, equivalent formations of zero forcing have been developed in different settings including sensor allocation to solve linear systems (K.-Lin 2018), controllability in follower-leader dynamics (Mon…
A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ without isolated vertices, then $S$ is called a total forcing set in $G$. The minimum cardinalit…
A global forcing set for maximal matchings of a graph $G=(V(G), E(G))$ is a set $S \subseteq E(G)$ such that $M_1\cap S \neq M_2 \cap S$ for each pair of maximal matchings $M_1$ and $M_2$ of $G$. The smallest such set is called a minimum global forcing set, its size being the global forcing number for maximal matchings $ϕ_{gm}(G)$ of $G$. In this paper, we establish lower and upper bounds on the forcing number for maximal matchings of the corona product of graphs. We also introduce an integer linear programming model for computing the forcing number for maximal matchings of graphs.
Let $S$ be a set of vertices of a graph $G$. Let $cl(S)$ be the set of vertices built from $S$, by iteratively applying the following propagation rule: if a vertex and all but exactly one of its neighbors are in $cl(S)$, then the remaining neighbor is also in $cl(S)$. A set $S$ is called a zero forcing set of $G$ if $cl(S)=V(G)$. The zero forcing number $Z(G)$ of $G$ is the minimum cardinality of a zero forcing set. Let $cl(N[S])$ be the set of vertices built from the closed neighborhood $N[S]$ of $S$, by iteratively applying the previous propagation rule. A set $S$ is called a power dominatin…
Let G be a graph with a perfect matching. A complete forcing set of G is a subset of edges of G to which the restriction of every perfect matching is a forcing set of it. The complete forcing number of G is the minimum cardinality of complete forcing sets of G. Xu et al. gave a characterization for a complete forcing set and derived some explicit formulas for the complete forcing numbers of cata-condensed hexagonal systems. In this paper, we consider general hexagonal systems. We present an upper bound on the complete forcing numbers of hexagonal systems in terms of elementary edge-cut cover a…
Harary et al. and Klein and Randic proposed the forcing number of a perfect matching in mathematics and chemistry, respectively. In detail, the forcing number of a perfect matching M of a graph G is the smallest cardinality of subsets of M that are contained in no other perfect matchings of G. The author and cooperators defined the forcing polynomial of G as the count polynomial for perfect matchings with the same forcing number of G, from which the average forcing number, forcing spectrum, and the maximum and minimum forcing numbers of G can be obtained. Up to now, a few papers have been cons…
Given a graph $G$ and a real number $0\le p\le 1$, we define the random set $B_p(G)\subset V(G)$ by including each vertex independently and with probability $p$. We investigate the probability that the random set $B_p(G)$ is a zero forcing set of $G$. In particular, we prove that for large $n$, this probability for trees is upper bounded by the corresponding probability for a path graph. Given a minimum degree condition, we also prove a conjecture of Boyer et.\ al.\ regarding the number of zero forcing sets of a given size that a graph can have.
