Secure Vertex Edge Domination of Some Middle Graphs

Abstract

Graph theory is one of the fields associated with contemporary mathematics and computer science that has been growing the fastest. The most rapidly evolving branch of graph theory is domination. When there is a restricted amount of facilities (which include fire stations and hospitals) and every attempt is made to shorten the distance a person must travel in order to get to the nearest company, accessibility issues become dominant. A comparable issue arises when one tries to enforce a certain maximum distance to a facility while minimizing the number of facilities available to service individuals. Domination notions are also spotted in mapping, communication or electrical network surveillance, and challenges related to creating delegations.  { J } [V (G)] is a secure vertex edge dominating set of G,  Assume each edge, e [E (G)],  a vertex exists,  V { J } such that V defends edge e. That is, the edges incident on a vertex in {J}together with the edges adjacent to the incident edgeare protected by that vertex.A secure vertex edge dominating { J }G has the characteristic of being the dominant set in a graph where each vertex  y [V –  { J }]  is adjacent to either a vertex , adjacent to a vertex as well an incident edge of y, x   { J }which means ({J} - {x}) {y} is a dominating set. The phrase "secure vertex edge domination number" corresponds to lowest cardinality of secure vertex edge domination in G and corresponds to the lowest cardinality of secure vertex edge domination in G and is represented by . We kick off the current studyby investigating the novel parameter and determining the secure vertex edge dominance number of particular middle graph kinds that are members of particular graph families, includingpath, cycles, complete, wheel and friendship graphs.