International Journal of Combinatorics

Volume 2013 (2013), Article ID 863439, 7 pages

http://dx.doi.org/10.1155/2013/863439

## Some New Classes of Open Distance-Pattern Uniform Graphs

Department of Mathematics, S.D. College Alappuzha, University of Kerala, Kerala 688003, India

Received 8 April 2013; Accepted 15 June 2013

Academic Editor: Toufik Mansour

Copyright © 2013 Bibin K. Jose. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Given an arbitrary nonempty subset of vertices in a graph , each vertex in is associated with the set and called its open -distance-pattern. The graph is called open distance-pattern uniform (odpu-) graph if there exists a subset of such that for all and is called an open distance-pattern uniform (odpu-) set of The minimum cardinality of an odpu-set in , if it exists, is called the odpu-number of and is denoted by . Given some property , we establish characterization of odpu-graph with property . In this paper, we characterize odpu-chordal graphs, and thereby characterize interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, and ptolemaic graphs that are odpu-graphs. We also characterize odpu-self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We prove that the odpu-number of cographs is even and establish that any graph can be embedded into a self-complementary odpu-graph , such that and are induced subgraphs of . We also prove that the odpu-number of a maximal outerplanar graph is either or .