Forcing sets in triangular grid networks
Jessy Sujana. G, T. M. Rajalaxmi · Procedia Computer Science · 2020
In a graph G , suppose S is a subset of vertices which are all colored and the rest of the vertices are not colored. The dynamic coloring of the vertices is defined as, at each discrete time interval, a colored vertex forces exactly one vertex which is not colored to be colored. This process continues to make all the vertices colored. The subset S is called a forcing set of G . The forcing number ζ ( G ) of a graph G is the minimum cardinality of a set S with colored vertices which forces the set V ( G ) to be colored after some time. If the subset S has an additional property that it induces a subgraph of G whose components are all edges, then S is called a ζP 2 -forcing set of G . The minimum cardinality of a P 2 -forcing set of G with a is called the P 2 -forcing number of G and is denoted by ζP 2 (G). Analogous to the P 2 -forcing set, we define set S as a P 3 -forcing set if all components of S are vertex disjoint paths on 3 vertices. The minimum cardinality of a P 3 -forcing set is called the P 3 -forcing number of G and is denoted by ζP 3 ( G ). We compute the P 2 -forcing number and P 3 -forcing number of the Triangular grid network.