The current study aims to establish a connection between graphs and automata theory, which apparently demonstrate di erent mathematical structures. Through searching out some properties of one of these structures, we try to  nd some new properties of the other structure as well. This will result in obtaining some unknown properties. At  rst, a novel automaton called zero-FORCING (Z-F)  nite automata is de ned according to the notion of a zero-FORCING set of a graph. It is shown that for a given graph and for some zero FORCING sets, various Z-F- nite automata will be obtained. In addition, the language and the closure properties of Z-F- nite automata, in particular; union, connection, and serial connection are studied. Moreover, considering some properties of graphs such as the closed trail, connected and complete; some new features for Z-F- nite automata are presented. Further, it is shown that there is not any  nite graph such that f be a part of the language of its Z-F- nite automata. Actually, it is proved that for every given graph, the Z-F- nite automata of it does not show any closed trail containing all edges for every zero FORCING set, but if the graph G has been a closed trail containing all edges, then the Z-F- nite automata of it has a weak closed trail containing all edges. Some examples are also given to clarify these new notions.

The idea of “FORCING” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as Latin squares, block designs and Steiner systems in combinatorics [D. Donovan, E. S. Mahmoodian, C. Ramsay, A. P. Street, Defining sets in combinatorics: A survey, in: C. D. Wensley (Ed.), Surveys in Combinatorics, Cambridge Univ. Press, 2003, pp. 115-174]. Recently, the FORCING on perfect matchings has been attracting more researchers’ attention. A FORCING set of a perfect matching M of a graph G is a subset of M contained in no other perfect matchings of G. A global FORCING set of G, introduced by Vukičvićet al., is a subset of E (G) on which there are distinct restrictions of any two different perfect matchings of G. Combining the above “FORCING” and “global” ideas. Xu et al. in [Complete FORCING numbers of catacondensed benzenoid, J. Combin. Optim.29 (2015) 803-814.] introduced a complete FORCING set of G defined as a subset of E (G) on which the restriction of any perfect matching M of G is a FORCING set of M. The minimum cardinality of complete FORCING sets is the complete FORCING number of G. In this paper, we give the explicit expressions for the complete FORCING number of several classes of polyphenyl systems.

A subset of the vertex set of a graph $G$ is called a zero FORCING set if by considering them colored and, as far as possible, a colored vertex with exactly one non-colored neighbor forces its non-colored neighbor to get colored, then the whole vertices of $G$ become colored. The total FORCING number of a graph $G$, denoted by $F_t(G)$, is the cardinality of a smallest zero FORCING set of $G$ which induces a subgraph with no isolated vertex. The connected FORCING number, denoted by $F_c(G)$, is the cardinality of a smallest zero FORCING set of $G$ which induces a connected subgraph. In this paper, we first characterize the graphs with $F_t(G)=2$ and, as a corollary, we characterize the graphs with $F_c(G)=2$.

Let be a simple connected graph. A perfect matching (or Kekul'e structure in chemical literature) of is a set of disjoint edges which covers all vertices of . The anti-FORCING number of 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 . For every , the th power of , denoted by , is a graph with the same vertex set as such that two vertices are adjacent in if and only if their distance is at most in . In this paper, we study the anti-FORCING number of the powers of some graphs.

‎For a connected graph $G=(V,E)$ of order at least two‎, ‎an edge detour monophonic set of $G$ is a set $S$ of vertices such that every edge of $G$ lies on a detour monophonic path joining some pair of vertices in $S$‎. ‎The edge detour monophonic number of $G$ is the minimum cardinality of its edge detour monophonic sets and is denoted by $edm(G)$‎. ‎A subset $T$ of $S$ is a FORCING edge detour monophonic subset for $S$ if $S$ is the unique edge detour monophonic set of size $edm(G)$ containing $T$‎. ‎A FORCING edge detour monophonic subset for $S$ of minimum cardinality is a minimum FORCING edge detour monophonic subset of $S$‎. ‎The FORCING edge detour monophonic number $f_{edm}(S)$ in $G$ is the cardinality of a minimum FORCING edge detour monophonic subset of $S$‎. ‎The FORCING edge detour monophonic number of $G$ is $f_{edm}(G)=min\{f_{edm}(S)\}$‎, ‎where the minimum is taken over all edge detour monophonic sets $S$ of size $edm(G)$ in $G$‎. ‎We determine bounds for it and find the FORCING edge detour monophonic number of certain classes of graphs‎. ‎It is shown that for every pair a‎, ‎b of positive integers with $0\leq a

In this paper, for a given graph, G, some domination parameters and the FORCING domination number of the graph, M (G), obtained from G arising in Mycielski's construction, are studied.

The zero FORCING number of a graph $G$, denoted $Z(G)$, is a graph parameter  which is based on a color change rule that describes how to color the vertices. Zero FORCING is useful in several branches of science such as electrical engineering, computational complexity and quantum control.  In this paper, we investigate the zero FORCING number for Cartesian products of some graphs. The main contribution of this paper is to introduce a new presentation of the Cartesian product of two complete bipartite graphs and to obtain the zero FORCING number of these graphs.  We also introduce a purely graph theoretical method to prove $Z(K_n \Box K_m)=mn-m-n+2$.

The zero FORCING number of a graph is the minimum cardinality among all the zero FORCING sets of a graph $G$.  The aim of this article is to compute the zero FORCING number of complementary prism graphs.  Some bounds on the zero FORCING number of complementary prism graphs are presented. The remainder of this article discusses the following result.  Let $G$ and $\overline{G }$ be connected graphs. Then $Z(G\overline{G})\leq n-1$ if and only if  there exists two vertices $v_i,v_j \in V(G)$ and $i\neq j$ such that, either $N(v_i) \subseteq N(v_j)$ or $N[v_i] \subseteq N[v_j]$ in $G$.