On the global forcing number of hexagonal systems

Global Forcing Number for Maximal Matchings under Graph Operations

Let $S= \{e_1,\,e_2‎, ‎\ldots,\,e_m\}$ be an ordered subset of edges of a connected graph $G$‎. ‎The edge $S$-representation of an edge set $M\subseteq E(G)$ with respect to $S$ is the‎ ‎vector $r_e(M|S) = (d_1,\,d_2,\ldots,\,d_m)$‎, ‎where $d_i=1$ if $e_i\in M$ and $d_i=0$‎ ‎otherwise‎, ‎for each $i\in\{1,\ldots‎ , ‎k\}$‎. ‎We say $S$ is a global forcing set for maximal matchings of $G$‎ ‎if $...

Hexagonal systems with forcing edges

An edge of a hexagonal system H is said to be forcing if it belongs to exactly one perfect matching of H. Using the concept of Z-transformation of hexagonal system, we give a characterization for the hexagonal systems with forcing edges and determine all forcing edges is such systems. We also give the generating function of all hexagonal systems with forcing edges. A hexaoonal system, also call...

On the zero forcing number of some Cayley graphs

‎Let Γa be a graph whose each vertex is colored either white or black‎. ‎If u is a black vertex of Γ such that exactly one neighbor‎ ‎v of u is white‎, ‎then u changes the color of v to black‎. ‎A zero forcing set for a Γ graph is a subset of vertices Zsubseteq V(Γ) such that‎ if initially the vertices in Z are colored black and the remaining vertices are colored white‎, ‎then Z changes the col...

Anti-forcing number of some specific graphs

Let $G=(V,E)$ be a simple connected graph. A perfect matching (or Kekul'e structure in chemical literature) of $G$ is a set of disjoint edges which covers all vertices of $G$. The anti-forcing number of $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by $af(G)$. In this paper we consider some specifi...

Global forcing number of grid graphs