Related papers: Relations between global forcing number and maximu…
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 forcing spectrum of $G$ is the collection of forcing numbers of all…
It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph. In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number…
In a graph G, a dominating set Df subset of V (G) is called a dom-forcing set if the sub-graph induced by Df must form a zero forcing set. The minimum cardinality of such a set is known as the dom-forcing number of the graph G, denoted by…
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…
In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected…
Let $G$ be a graph with a perfect matching. Denote by $f(G)$ the minimum size of a matching in $G$ which is uniquely extendable to a perfect matching in $G$. Diwan (2019) proved by linear algebra that for $d$-hypercube $Q_d$ ($d\geq 2)$,…
Given a simple undirected graph $G$ and a positive integer $k$, the $k$-forcing number of $G$, denoted $F_k(G)$, is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the…
While a number of bounds are known on the zero forcing number $Z(G)$ of a graph $G$ expressed in terms of the order of a graph and maximum or minimum degree, we present two bounds that are related to the (upper) total domination number…
The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $F\subseteq V(G)$ such that only the trivial automorphism of $G$ fixes every vertex in $F$. Let $\Pi$ $=$ $\{F_1,F_2,\ldots,F_k\}$ be an ordered $k$-partition…
Motivated by a conjecture from the automated conjecturing program TxGraffiti, in this paper the relationship between the zero forcing number, $Z(G)$, and the vertex independence number, $\alpha(G)$, of cubic and subcubic graphs is explored.…
The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph,…
A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most $1$. Computing the dissociation number ${\rm diss}(G)$ of a given graph $G$, defined as the order of a maximum dissociation set in $G$, is…
The general position number ${\rm gp}(G)$ of a graph $G$ is the cardinality of a largest set of vertices $S$ such that no element of $S$ lies on a geodesic between two other elements of $S$. The complementary prism $G\overline{G}$ of $G$ is…
The positive zero forcing number of a graph is a graph parameter that arises from a non-traditional type of graph colouring, and is related to a more conventional version of zero forcing. We establish a relation between the zero forcing and…
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…
Let $G$ be a graph, and $Z$ a subset of its vertices, which we color black, while the remaining are colored white. We define the skew color change rule as follows: if $u$ is a vertex of $G$, and exactly one of its neighbors $v$, is white,…
The forcing number of a perfect matching of a graph was introduced by Harary et al., which originated from Klein and Randi\'{c}'s ideal of innate degree of freedom of Kekul\'{e} structure in molecular graph. On the opposite side in some…
In this paper, we study minimal (with respect to inclusion) zero forcing sets. We first investigate when a graph can have polynomially or exponentially many distinct minimal zero forcing sets. We also study the maximum size of a minimal…
Zero forcing is a dynamic graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. This forcing process has been used to approximate certain linear algebraic parameters, as well as…
This article studies the \emph{$k$-forcing number} for oriented graphs, generalizing both the \emph{zero forcing number} for directed graphs and the $k$-forcing number for simple graphs. In particular, given a simple graph $G$, we introduce…