# Planar graph characterization of NDSS graphs

01 Nov 2017-Vol. 263, Iss: 4, pp 042129

TL;DR: This paper obtains a necessary and sufficient condition for a graph to be NDSS and hence characterize the planarity and outer – planarity of its complement.

Abstract: Planar graph characterization is always of interest due to its complexity in characterization. In this paper, we obtain a necessary and sufficient condition for a graph to be NDSS and hence characterize the planarity and outer – planarity of its complement .

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VIT University

^{1}TL;DR: A uniquely colorable graph G whose chromatic partition contains atleast one g-set is termed as a g-uniquely-coloring graph as discussed by the authors, and the necessary and sufficient condition for and G* to be g -uniquely colorable whenever G G is g-colorable such that |P | ³ 2, G can be both planar and nonplanar.

Abstract: A uniquely colorable graph G whose chromatic partition contains atleast one g - set is termed as a g - uniquely colorable graph. In this paper, we provide necessary and sufficient condition for and G* to be g - uniquely colorable whenever G g- uniquely colorable and also provide constructive characterization to show that whenever G is g- uniquely colorable such that |P | ³ 2, G can be both planarand non planar.

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VIT University

^{1}TL;DR: In this article, a uniquely colorable graph G whose chromatic partition contains at least one γ-set is characterized as a γ uniquely colourable graph and characterized the planarity of these graphs using the domination number of G.

Abstract: A uniquely colorable graph G whose chromatic partition contains at least one γ-set is termed as a γ-uniquely colorable graph. We characterize the planarity of these graphs using the domination number of G.

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TL;DR: In this paper, the conditions under which both a graph G and its complement G¯ possess a specified property were investigated, and all graphs G for which G and G& #175 have connectivity one, (b) have line-connectivity one, and (c) are 2-connected, (d) are forests, (e) are bipartite, (f) are outerplanar and (g) are eulerian.

Abstract: We investigate the conditions under which both a graph G and its complement G¯ possess a specified property. In particular, we characterize all graphs G for which G and G¯ both (a) have connectivity one, (b) have line-connectivity one, (c) are 2-connected, (d) are forests, (e) are bipartite, (f) are outerplanar and (g) are eulerian. The proofs are elementary but amusing.

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