A Textbook of Graph Theory

Abstract

Vertex-independent sets and vertex coverings as also edge-independent sets and edge coverings of graphs occur very naturally in many practical situations and hence have several potential applications. In this chapter, we study the properties of these sets. In addition, we discuss matchings in graphs and, in particular, in bipartite graphs. Matchings in bipartite graphs have varied applications in operations research. We also present two celebrated theorems of graph theory, namely, Tutte's 1-factor theorem and Hall's matching theorem. All graphs considered in this chapter are loopless. Vertex-Independent Sets and Vertex CoveringsDefinition 5.2.1. A subset S of the vertex set V of a graph G is called independent if no two vertices of S are adjacent in G. S Â V is a maximum independent set of G if G has no independent set S 0 with jS 0 j > jS j. A maximal independent set of G is an independent set that is not a proper subset of another independent set of G.For example, in the graph of Fig. 5.1, fu; v; wg is a maximum independent set and fx; yg is a maximal independent set that is not maximum. Definition 5.2.2.A subset K of V is called a covering of G if every edge of G is incident with at least one vertex of K. A covering K is minimum if there is no covering K 0 of G such that jK 0 j < jKjI it is minimal if there is no covering K 1 of G such that K 1 is a proper subset of K.In the graph W 5 of Fig. 5.2, fv 1 ; v 2 ; v 3 ; v 4 ; v 5 g is a covering of W 5 and fv 1 ; v 3 ; v 4 ; v 6 g is a minimal covering. Also, the set fx; yg is a minimum covering of the graph of Fig. 5.1.

NLS ground states on graphs

“…Although we shall not need deep results from graph theory, the notion of graph is central to this paper: we refer the reader to [5,7] for a modern account on the subject.…”

Diffusion and reaction pathways of water near fully hydrated TiO2 surfaces from ab initio molecular dynamics

“…Labels created from the elements of the graph identify molecules based on indistinguishable atoms without considering stereochemistry. Labels created from isomorphic graphs 96 identify molecules based on indistinguishable atoms and stereochemistry. We used unique labels, l = V n : p n , that were created from a sorted dictionary with the vertex labels for keys and a list of the p n neighbor vertices to n (with n and p n sorted in ascending order) for values.…”

On an edge partition and root graphs of some classes of line graphs

“…First, we show that the sequence (a 1 , a 2 , 2, a 3 , a 4 ), where a i ∈ {1, 2}, is forbidden in L * . For, assuming the contrary, let 4 , v 5 and v 6 be arbitrary vertices in L j , for j = i − 2, i − 1, i + 1 and i + 2 respectively. But < v 1 , .…”

“…A graph G is chordal if every cycle of length at least four in G has an edge(chord) joining two non-adjacent vertices of the cycle [4]. A graph is Ptolemaic if it is both distance-hereditary and chordal [11].…”