Distance domination in graphs with given minimum and maximum degree

Let G be a connected graph and k∈double-struckN. The k‐distance domination number of G is the smallest cardinality of a set S of vertices such that every vertex of G is within distance k from some vertex of S. While for k=1, that is, for the ordinary domination number, the problem of finding asymptotically sharp upper bounds in terms of order and minimum degree of the graph has been solved, corresponding bounds for k>1 have remained elusive. In this paper, we solve this problem and present an asymptotically sharp upper bound on the k‐distance domination number of a graph in terms of its order and minimum degree, which significantly improves on bounds in the literature. We also obtain an asymptotically sharp upper bound on the p‐radius of graphs in terms of order and minimum degree. For p∈double-struckN, the p‐radius of G is defined as the smallest integer d such that there exists a set S of p vertices of G having the property that every vertex of G is within distance d of some vertex in S. We also present improved bounds for graphs of given order, minimum degree and maximum degree, for triangle‐free graphs and for graphs not containing a 4‐cycle as a subgraph.

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“…These bounds were improved in the paper by Henning and Lichiardopol , who established the following results.…”

Section: Introductionmentioning

confidence: 94%

“…k Theorem 6 (Henning and Lichiardopol [16]). For ≥ k 2, if G is a connected graph with a minimum degree ≥ δ 2 and maximum degree Δ and of order ≥…”

Section: Introductionmentioning

confidence: 99%

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Distance domination and generalized eccentricity in graphs with given minimum degree

Dankelmann

Erwin

2019

Journal of Graph Theory

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“…Kundu and Majumder [16] and Barman et al [3] presented an optimal linear-time algorithm for the problem on trees and interval graphs, respectively. The k-dominating number of a graph G, denoted by γ k (G), is the cardinality of a minimum k-hop dominating set of G. The k-dominating number has been widely studied in the literature; see for example [7,10,12,18,21].…”

Section: Introductionmentioning

confidence: 99%

A Linear-Time Algorithm for Minimum $k$-Hop Dominating Set of a Cactus Graph

Abu-Affash¹,

Carmi²,

Krasin³

2020

Preprint

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Given a graph G = (V, E) and an integer k ≥ 1, a k-hop dominating set D of G is a subset of V , such that, for every vertex v ∈ V , there exists a node u ∈ D whose hop-distance from v is at most k. A k-hop dominating set of minimum cardinality is called a minimum k-hop dominating set. In this paper, we present linear-time algorithms that find a minimum k-hop dominating set in unicyclic and cactus graphs. To achieve this, we show that the k-dominating set problem on unicycle graph reduces to the piercing circular arcs problem, and show a linear-time algorithm for piercing sorted circular arcs, which improves the best known O(n log n)-time algorithm.

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“…The modern study of domination has connections to game theory, coding theory, and matching theory; see [13] and [12] for extensive background. The extension of domination notions to the cases d > 1 in Definitions 1.1 and 1.2 follows [19] and [14], for example. The case d = 1 of Definition 1.3 corresponds to the γ-graph γ • G, introduced by Subramanian and Sridharan [29] and subsequently studied in [28], [1], [27], [3].…”

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confidence: 99%

Which graphs occur as $γ$-graphs?

DeVos¹,

Dyck²,

Jedwab³

et al. 2018

Preprint

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The γ-graph of a graph G is the graph whose vertices are labelled by the minimum dominating sets of G, in which two vertices are adjacent when their corresponding minimum dominating sets (each of size γ(G)) intersect in a set of size γ(G) − 1. We extend the notion of a γ-graph from distance-1-domination to distance-d-domination, and ask which graphs H occur as γgraphs for a given value of d ≥ 1. We show that, for all d, the answer depends only on whether the vertices of H admit a labelling consistent with the adjacency condition for a conventional γ-graph. This result relies on an explicit construction for a graph having an arbitrary prescribed set of minimum distance-d-dominating sets. We then completely determine the graphs that admit such a labelling among the wheel graphs, the fan graphs, and the graphs on at most six vertices. We connect the question of whether a graph admits such a labelling with previous work on induced subgraphs of Johnson graphs. IntroductionIn this paper we consider only finite, loop-free, undirected graphs G without multiple edges. Our main object of study is the γ d -graph of a graph G, which we introduce via the following three definitions.Definition 1.1. Let G be a graph, and let S and T be subsets of the vertex set V (G) of G. The set S distance-d-dominates T if every vertex of T is within distance d in G of some vertex in S. In the case T = V (G), the subset S is a distance-d-dominating set of G. Definition 1.2. A minimum distance-d-dominating set of a graph G is a distance-d-dominating set of smallest size, and this size is the distance-d-domination number γ d (G) of G.These definitions reduce to well-studied domination notions when d = 1: a distance-1-dominating set is a dominating set; a minimum distance-1-dominating set is a minimum dominating set; and the distance-1-domination number γ 1 (G) is the domination number γ(G). The study of domination in graphs spans more than fifty years, with early interpretations that include the number of queens

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Distance domination in graphs with given minimum and maximum degree

Cited by 11 publications

References 7 publications

Distance domination and generalized eccentricity in graphs with given minimum degree

Distance domination and generalized eccentricity in graphs with given minimum degree

A Linear-Time Algorithm for Minimum $k$-Hop Dominating Set of a Cactus Graph

Which graphs occur as $γ$-graphs?

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