Combinatorial Algorithms

A graph G = (V,E) is γ-quasi-complete (γ ∈ [0, 1]) if every vertex in G is connected to at least γ.(|V | − 1) other vertices. In this paper, we establish some relationships between the girth and the quasi-completeness of a graph. We also derive an upper bound 1 2 ( 1 + r γ ) + √ 1 4 ( 1 + r γ )2 + 2|E| γ − r|V | γ for the largest order γ-quasi-complete subgraph in a graph of minimum degree r. 1 Basic Definitions and Preliminaries Throughout this paper, the term graph is used to denote an unweighted and undirected simple graph (without self-loops or parallel edges) G = (V,E), where V and E are the vertex and edge sets, respectively [5]. The degree of a vertex v, denoted as d(v) in a graph, is the number of edges incident to it. A graph is called r-regular if every vertex in the graph has degree r. Let us denote δ(G) = min∀v∈V d(v), the minimum degree of a graph G. The order and size defines the cardinality of the vertex set and the edge set of a graph, respectively. A subgraph G′ = (V ′, E′) of a graph G (G′ ⊆ G) is defined such that V ′ ⊆ V and E′ ⊆ E. Let us assume that the cardinality of a set S is represented as |S|. Now, we have the following basic definitions. Definition 1 (γ-quasi-complete graph). A connected graph, G = (V,E), is said to be γ-quasi-complete (0 < γ ≤ 1) if every vertex in the graph has degree value at least γ.(|V | − 1), i.e., δ(G) ≥ γ.(|V | − 1). Definition 2 (γ-quasi-clique). In a graph G = (V,E), a subset of vertices V ′ ⊂ V forms a γ-quasi-clique (0 < γ ≤ 1) if the subgraph induced by V ′, GV ′ , is a γ-quasi-complete graph. Fig. 1(a) shows an example of a quasi-clique in a quasi-complete graph. There are some other versions of the problem for an n vertex subgraph with m edges by the terminology ‘γ-clique’, where m ≥ γ.n.(n−1) 2 [1] or ‘γ-near-clique’, where m ≥ (1−γ).n.(n−1) 2 [7], which are however different by definition. Quasi-cliques are generalization of the concept of a clique [6], and therefore, the problem of finding the maximum quasi-clique is computationally harder than the maximum clique problem. Note that, a γ-quasi-clique in any arbitrary graph G is always a S. Arumugam and B. Smyth (Eds.): IWOCA 2012, LNCS 7643, pp. 1–5, 2012. c © Springer-Verlag Berlin Heidelberg 2012 2 M. Bhattacharyya and S. Bandyopadhyay