INDUCED SUBGRAPHS

Abstract

This chapter elaborates the different aspects of induced subgraphs. It is assumed that k is a nonnegative integer and G is a graph. A subgraph H of G is said to be induced if every edge of G joining vertices of H is in H. By a k-subgraph of G, one means an induced subgraph of order k of G. The aim is to study graphs whose k-subgraphs fulfill certain conditions. The theorems are applied to get results concerning three graph invariants which include the inducibility, the vitality, and the modality of a graph. All graphs considered are finite and undirected. Loops and multiple edges are allowed. A graph is said to be loopless if it has no loops (multiple edges, respectively). A loopless simple graph is called ordinary. A graph that is not loopless [simple] is said to be a pseudograph [a multigraph, respectively]. A graph G is said to be quasicomplete if every vertex of G has the same number of loops and every two distinct vertices of G are joined by the same number of edges.