Hamiltonicity Of Transformation Graph G^+--

All graphs considered here are finite, undirected and simple. The total graph T(G) ofG is the graph whose vertex set is V (G)∪E(G), and in which two vertices are adjacentif and only if they are adjacent or incident in G. Wu and Meng introduced some newgraphical transformations which generalize the concept of total graph. In this paper, weshall investigate the transformation graph G+?? of a graph G. The transformation graphG+?? of a graph is the graph with vertex set V (G)∪E(G), in which two vertices u and vare joined by an edge if one of the following conditions holds: (i) u,v∈V (G) and they areadjacent in G, (ii) u,v∈E(G) and they are not adjacent in G, (iii) one of u and v is inV (G) while the other is in E(G), and they are not incident in G. In 1975, Fleischner andHobbs showed that G+++ is hamiltonian if and only if G contains an EPS-subgraph, thatis, a connected spanning subgraph S which is the edge-disjoint union of a (not necessarilyconnected) graph E, all of whose vertices have even degree, with a (possibly empty) forestP each of whose component is a path. In this paper, for any graph G, we determine theindependence number and the connectivity of G+??. Furthermore, we show that for agraph G with no isolated vertices, G+?? is hamiltonian if and only if G is neither a star,nor G∈{2K2,K3,K1,1}.