Uniformly dissociated graphs

Brešar, Boštjan; Hartnell, Bert L.; Rall, Douglas F.

doi:10.26493/1855-3974.1013.46a

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…To conclude the section we are going to show that the geodesic-transversal problem is NP-complete. In the study of vertex-deletion problems [47], the concept of a dissociation set (see [4,22,40]) was considered, which was shown in [6] to be the complement of a 3-path vertex cover in any graph. Since dissociation set problem is NP-complete even when restricted to bipartite graphs [47], we infer the following.…”

Section: Basic Observations and Np-completenessmentioning

confidence: 99%

The geodesic-transversal problem

Manuel¹,

Brešar²,

Klavžar³

2021

Preprint

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A maximal geodesic in a graph is a geodesic (alias shortest path) which is not a subpath of a longer geodesic. The geodesic-transversal problem in a graph G is introduced as the task to find a smallest set S of vertices of G such that each maximal geodesic has at least one vertex in S. The minimum cardinality of such a set is the geodesic-transversal number gt(G) of G. It is proved that gt(G) = 1 if and only if G is a subdivided star and that the geodesic-transversal problem is NP-complete. Fast algorithms to determine the geodesic-transversal number of trees and of spread cactus graphs are designed, respectively.

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Section: Basic Observations and Np-completenessmentioning

confidence: 99%

The geodesic-transversal problem

Manuel¹,

Brešar²,

Klavžar³

2021

Preprint

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“…If G is a graph and S ⊆ V (G), then S is a dissociation set if ⟨S⟩ has maximum degree at most 1. The dissociation number diss(G) of G is the cardinality of a largest dissociation set in G. This concept was introduced by Yanakkakis [22]; see also [3,4,9]. Further, a k-path vertex cover of G is a subset S of vertices of G such that every path of order k in G contains at least one vertex from S. The minimum cardinality of a k-path vertex cover in G is denoted by ψ k (G).…”

Section: Dissociation Number and Independence Numbermentioning

confidence: 99%

General $d$-position sets

Klavzar,

Rall,

Yero

2020

Preprint

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The general d-position number gp d (G) of a graph G is the cardinality of a largest set S for which no three distinct vertices from S lie on a common geodesic of length at most d. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of gp d (G) with respect to the suitable values of d. We show that the decision problem concerning finding gp d (G) is NP-complete for any value of d. The value of gp d (G) when G is a path or a cycle is computed and a structural characterization of general d-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that gp d (G) is infinite whenever G is an infinite graph and d is a finite integer.

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“…[6,16]). We mention in passing that graphs in which all maximal dissociation sets are of the same size were studied in [3].…”

Section: Introductionmentioning

confidence: 99%

On the dissociation number of Kneser graphs

Brešar,

Dravec

2021

Preprint

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A set D of vertices of a graph G is a dissociation set if each vertex of D has at most one neighbor in D. The dissociation number of G, diss(G), is the cardinality of a maximum dissociation set in a graph G. In this paper we study dissociation in the well-known class of Kneser graphs K n,k . In particular, we establish that the dissociation number of Kneser graphs Kn,2 equals max {n − 1, 6}. We show that for any k ≥ 2, there exists n0 ∈ N such that diss(K n,k ) = α(K n,k ) for any n ≥ n0. We consider the case k = 3 in more details and prove that n0 = 8 in this case. Then we improve a trivial upper bound 2α(K n,k ) for the dissociation number of Kneser graphs K n,k by using Katona's cyclic arrangement of integers from {1, . . . , n}. Finally we investigate the odd graphs, that is, the Kneser graphs with n = 2k + 1. We prove that diss(K 2k+1,k ) = 2k k .

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Uniformly dissociated graphs

Cited by 4 publications

References 15 publications

The geodesic-transversal problem

The geodesic-transversal problem

General $d$-position sets

On the dissociation number of Kneser graphs

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