OPEN PACKING NUMBER FOR SOME CLASSES OF PERFECT GRAPHS

Kameshwar Chandrasekar ,S Saravanakumar

doi:10.15826/umj.2020.2.004

Abstract

Let \(G\) be a graph with the vertex set \(V(G)\). A subset \(S\) of \(V(G)\) is an open packing set of \(G\) if every pair of vertices in \(S\) has no common neighbor in \(G.\) The maximum cardinality of an open packing set of \(G\) is the open packing number of \(G\) and it is denoted by \(\rho^o(G)\). In this paper, the exact values of the open packing numbers for some classes of perfect graphs, such as split graphs, \(\{P_4, C_4\}\)-free graphs, the complement of a bipartite graph, the trestled graph of a perfect graph are obtained.

Highlights

By a graph G = (V, E), we mean a finite, undirected graph with neither loops nor multiple edges
For a subset S of V (G), the open and closed neighborhoods of S are defined by N (S) = ∪v∈SN (v) and N [S] = ∪v∈SN [v]
Uy ∈ E(G) and vx ∈ E(G) and the induced subgraph {u, v, x, y} is isomorphic to either P4 or C4 depending on xy ∈ E(G) or xy ∈ E(G), which is a contradiction to G is a {P4, C4}-free graph

Summary

Introduction

A subset S of V (G) is an open packing set of G if every pair of vertices in S has no common neighbor in G. Uy ∈ E(G) and vx ∈ E(G) and the induced subgraph {u, v, x, y} is isomorphic to either P4 or C4 depending on xy ∈ E(G) or xy ∈ E(G), which is a contradiction to G is a {P4, C4}-free graph. Any open packing set of G contains at most one vertex in N (x) = V (G) \ {x}, it follows that ρo(G) = 2. Since no two vertices in S have a common neighbor in G and the induced subgraph K is complete, it follows that |S ∩ K| ≤ 2.

Complement of a bipartite graph and the trestled graph of a graph

Conclusion and scope