The P3 intersection graph

Abstract

We define a new graph operator called the P₃ intersection graph, P₃(G)- the intersection graph of all induced 3-paths in G. A characterization of graphs G for which P₃ (G) is bipartite is given. Forbidden subgraph characterization for P₃(G) having properties of being chordal, H-free, complete are also obtained. For integers a and b with a > 1 and b ≥ a - 1, it is shown that there exists a graph G such that x(G) = a, x(P₃(G)) = b, where x is the chromatic number of G. For the domination number γ(G), we construct graphs G such that γ(G) = a and γ(P₃(G)) = b for any two positive numbers o > 1 and b. Similar construction for the independence number and radius, diameter relations are also discussed.