EDGE BURNING: RELATIONSHIP BETWEEN EDGE AND VERTEX BURNING OF GRID GRAPH

Consider a graph G with the Cartesian product of two Path graphs.In this paper we are concentrating on edge burning for specific Grid graph.The notation of edge burning number [7] of an m n Cartesian grid graph, denoted as b 1 (Gm,n), the vertex burning number of a grid b(Gm,n) graph was first studied in [9].In this paper, we focus exclusively on undirected graphs.Determining the exact edge burning number for an arbitrary grid graph is an NPhard problem.Burning is a discrete step by step process; the main intention of burning is to acquire a burning number within the limited period of time which is the so-called source of the vertex and We know that a burning number is denoted as b(G).Similarly edge burning is a source of the edge; by using the source edges we can completely burn a graph.The Grid graph is formed by taking the Cartesian product of two Path graphs, denoted by as .In this paper the main focus is on this graph.Specifically, we studied the edge burning number of the Cartesian product of two path graphs, even in non-symmetric cases (Where the two path graphs have different lengths.)Remarkably, we successfully demonstrated the following cases. When m < , the vertex burning number b(Gm,n).When m < , the edge burning number b 1 (Gm,n).When m = , the vertex burning number b (Gm,n).When m = , the edge burning number b 1 (Gm,n).When m > , the vertex burning number b (Gm,n).When m > , the edge burning number b 1 (Gm,n).And we are studying among the relationship between edge and vertex burning for the above cases.As well as we are giving the upper bounds for the Edge burning for Grid graph.Edge and vertex burning play crucial roles in ensuring connectivity and resilience within grid graphs.The relationship between these burning numbers provides valuable insights into the structure of such graphs.