On oriented cliques with respect to push operation

Bensmail, Julien; Nandi, Soumen; Sen, Sagnik

doi:10.1016/j.dam.2017.07.037

Cited by 9 publications

(16 citation statements)

References 13 publications

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“…The proof of the above result is correct in the paper [1] except for some errors in Lemma 5.9 from [1]. The statement of the lemma is unchanged nevertheless.…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

“…However, our list had an error. The graph − → H 10 , as illustrated in Figure 1 of our paper [1] is not a push clique. After finding the error, we rechecked our proof and found Figure 1: A list of planar push cliques whose underlying graphs H 1 , H 2 , · · · , H 16 is an exhaustive list of edge-minimal planar underlying push cliques.…”

Section: Introduction and Main Resultsmentioning

confidence: 90%

“…The theorem characterizes all minimal (up to spanning subgraph inclusion) underlying planar push cliques. The theorem claimed that there are exactly 16 such graphs and we provided a figure (Figure 1 in our paper [1]) which illustrated all the 16 graphs. In the figure, we actually presented orientations of those 16 undirected graphs in such a way that they were push cliques.…”

Section: Introduction and Main Resultsmentioning

confidence: 90%

“…This is an erratum reporting and correcting an error in our published article titled "On oriented cliques with respect to push operation" [1]. We spotted the error after the article was published.…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

“…The error occured in Theorem 1.3 of the concerned article [1]. The theorem characterizes all minimal (up to spanning subgraph inclusion) underlying planar push cliques.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On oriented cliques with respect to push operation

Bensmail

Nandi

Sen

2017

Discrete Applied Mathematics

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International audienceAn oriented graph is a directed graph without any directed cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. To push a vertex v of a directed graph G is to change the orientations of all the arcs incident to v. A push clique is an oriented clique that remains an oriented clique even if one pushes any set of vertices of it. We show that it is NP-complete to decide if an undirected graph is the underlying graph of a push clique or not. We also prove that a planar push clique can have at most 8 vertices and provide an exhaustive list of planar push cliques

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“…The proof of the above result is correct in the paper [1] except for some errors in Lemma 5.9 from [1]. The statement of the lemma is unchanged nevertheless.…”

Section: Introduction and Main Resultsmentioning

confidence: 87%

Section: Introduction and Main Resultsmentioning

confidence: 90%

Section: Introduction and Main Resultsmentioning

confidence: 90%

Section: Introduction and Main Resultsmentioning

confidence: 95%

“…The error occured in Theorem 1.3 of the concerned article [1]. The theorem characterizes all minimal (up to spanning subgraph inclusion) underlying planar push cliques.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On oriented cliques with respect to push operation

Bensmail

Nandi

Sen

2017

Discrete Applied Mathematics

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On Relative Clique Number of Triangle-Free Planar Colored Mixed Graphs

Nandi¹,

Sen

Taruni

2022

Lecture Notes in Computer Science

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An (n, m)-graph is a graph with n types of arcs and m types of edges. A homomorphism of an (n, m)-graph G to another (n, m)-graph H is a vertex mapping that preserves the adjacencies along with their types and directions. The order of a smallest (with respect to the number of vertices) such H is the (n, m)-chromatic number of G. Moreover, an (n, m)-relative clique R of an (n, m)-graph G is a vertex subset of G for which no two distinct vertices of R get identified under any homomorphism of G. The (n, m)-relative clique number of G, denoted by ω r(n,m) (G), is the maximum |R| such that R is an (n, m)-relative clique of G. In practice, (n, m)relative cliques are often used for establishing lower bounds of (n, m)-chromatic number of graph families.Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen [Discrete Applied Mathematics 2022] conjectured that ω r(n,m) (G) ≤ 2(2n + m) 2 + 2 for any triangle-free planar (n, m)-graph G and that this bound is tight for all (n, m) = (0, 1). In this article, we positively settle this conjecture by improving the previous upper bound of ω r(n,m) (G) ≤ 14(2n+m) 2 +2 to ω r(n,m) (G) ≤ 2(2n+m) 2 +2, and by finding examples of triangle-free planar graphs that achieve this bound. As a consequence of the tightness proof, we also establish a new lower bound of 2(2n + m) 2 + 2 for the (n, m)-chromatic number for the family of triangle-free planar graphs.

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