On cd-Coloring of Trees and Co-bipartite Graphs

Shalu, M. A.; Kirubakaran, V. K.

doi:10.1007/978-3-030-67899-9_16

Cited by 3 publications

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“…In the case of open packing, Henning and Slater [1] initiated the study on open packing and proved several bounds on ρ o (G) in trees and several other graph classes [1,6,7]. Rall [8] showed that for every non-trivial tree T , γ t (T ) = ρ o (T ) and in a recent work [9] (yet to be published), we extended this result by proving that the total domination number and the open packing number are equal when the underlying graph is a chordal bipartite graph with no isolated vertices. The decision version of the problem open packing is defined as follows: given a graph G and a positive integer k, the problem Open Packing checks whether G has an open packing of size at least k. It is known that Open Packing is NP-complete for bipartite graphs [9,10] and split graphs (a subclass of chordal graphs) [11].…”

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confidence: 99%

Open Packing in Interval Graphs

Shalu,

Kirubakaran

2024

Preprint

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Total Domination and Open Packing forms a primal-dual pair of problems. A vertex subset S of a graph G is called an open packing in G if no pair of distinct vertices in S have a common neighbour in G. The cardinality of a maximum open packing in G is called the open packing number, ρᵒ(G), of G which is a lower bound for the total domination number of G. Given a graph G and a positive integer k, the problem OPEN PACKING tests whether G has an open packing of size at least k. It is known that OPEN PACKING is NP-complete for split graphs (and so for chordal graphs) [Ramos et al., 2014]. In this work, we use a dynamic programming based approach to show that a maximum open packing in interval graphs (a subclass of chordal graphs) can be found in O(n³) time.

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