ON THE GENERALIZED xyz-LINE CUT TRANSFORMATION GRAPHS

Given a graph G with vertex set V(G), edge set E(G) and cutvertex set W(G), let ¯G be the complement, L(G) the line graph and C(G) the cutvertex graph of G. Let G^0 be the graph with V(G^0)=V(G) and without edges, G^1 the complete graph with vertex set V(G), G^+=G and G^-=¯G. Let lc(G) (¯lc(G)) be the graph whose vertices can be put in one to one correspondence with the set of edges and cutvertices of G in such a way that two vertices of lc(G) (resp.,¯lc(G)) are adjacent if and only if one corresponds to an edge of G and other to a cutvertex and they are incident (resp., nonincident). Given three variables x,y,z∈{0,1,+,-}, the generalized xyz-line cut transformation graph R^xyz (G) of G is graph with vertex set V(R^xyz (G))=E(G)∪W(G) and edge set E(R^xyz (G))=E(L(G))^x∪E(C(G))^y∪E(H), where H=lc(G) if z=+, H=¯lc(G) if z=-, H is the graph with V(H)=E(G)∪W(G) and without edges if z=0 and H is the complete bipartite graph with parts E(G) and W(G) if z=1. The graph R^xyz (G) generalizes the definition of the graph G^xz when y=0 and {x,z}⊆{+,-}, which is given in [1]. In this paper, we investigate some basic properties such as order, size, degree of a vertex and connectedness of generalized xyz-line cut transformation graphs.