Graphs &amp; Digraphs

Chartrand, Gary; Lesniak, Linda; Zhang, Ping

doi:10.1201/b19731

Cited by 458 publications

(366 citation statements)

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“…A is a κ-dominating set (or a dominating set when κ is understood) for (X, κ) if for every x ∈ X there exists a ∈ A such that x κ a [11]. This notion is somewhat analogous to that of a dense set in a topological space, and the following is somewhat analogous to the fact that in topological spaces, a continuous function is uniquely determined by its values on a dense subset of the domain.…”

Section: Definition and Basic Propertiesmentioning

confidence: 99%

Fixed point sets in digital topology, 2

Boxer

2020

Appl. Gen. Topol.

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Section: Definition and Basic Propertiesmentioning

confidence: 99%

Fixed point sets in digital topology, 2

Boxer

2020

Appl. Gen. Topol.

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“…The diameter of G is diam(G) = max{d(v, w) : v, w ∈ V (G)}. The independence number of G, denoted by α(G), is the maximum cardinality of an independent set of G. For undefined terminology, we refer the reader to [13].…”

Section: Introductionmentioning

confidence: 99%

Neighbor-locating colorings in graphs

Alcón

Gutiérrez

Hernando

et al. 2020

Theoretical Computer Science

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A k-coloring of a graph G is a k-partition Π = {S 1 , . . . , S k } of V (G) into independent sets, called colors. A k-coloring is called neighbor-locating if for every pair of vertices u, v belonging to the same color S i , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v. The neighbor-locating chromatic number χ N L (G) is the minimum cardinality of a neighbor-locating coloring of G.We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n ≥ 5 with neighbor-locating chromatic number n or n − 1. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphs.Henning and O. R. Oellermann introduced the so-called metric-locating-dominating sets, by merging the concepts of metric-locating set and dominating set.In [14], G. Chartrand, E. Salehi and P. Zhang, brought the notion of metric location to the ambit of vertex partitions, introducing the resolving partitions, also called metriclocating partition, and defining the partition dimension. Metric location and domination, in the context of vertex partitions, are studied in [28]. In [11], there were introduced the so-called locating colorings considering resolving partitions formed by independents sets.Neighbor location in sets was introduced by P. Slater in [39]. Given a graph G, a set S ⊆ V (G) is a dominating set if every vertex not in S is adjacent to some vertex in S. A set S ⊆ V (G) is a locating-dominating set if S is a dominating set and N (u) ∩ S = N (v) ∩ S for every two different vertices u and v not in S. The location-domination number of G, denoted by λ(G), is the minimum cardinality of a locating-dominating set. In [8,27], bounds for this parameter are given. In this paper, merging the concepts studied in [11,39], we introduce the neighbor-locating colorings and the neighbor-locating chromatic number, and examine this parameter in some families of graphs.

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“…In fact the above problems are related to safe set problems on weighted graphs. We use [2] for graph terminology and notation not defined here. Only finite, simple (undirected) graphs are considered.…”

Section: Introductionmentioning

confidence: 99%

On the weighted safe set problem on paths and cycles

et al. 2018

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Let G be a graph, and let w be a positive real-valued weight function on V (G). For every subset X of V (G), let w(X) = v∈X w(v). A non-empty subset S ⊂ V (G) is a weighted safe set of (G, w) if, for every component C of the subgraph induced by S and every component D of G − S, we have w(C) ≥ w(D) whenever there is an edge between C and D. If the subgraph of G induced by a weighted safe set S is connected, then the set S is called a connected weighted safe set of (G, w). The weighted safe number s(G, w) and connected weighted safe number cs(G, w) of (G, w) are the minimum weights w(S) among all weighted safe sets and all connected weighted safe sets of (G, w), respectively. It is easy to see that for any pair (G, w), s(G, w) ≤ cs(G, w) by their definitions. In this paper, we discuss the possible equality when G is a path or a cycle. We also give an answer to a problem due to Tittmann et al. [Eur. J. Combin. Vol. 32 (2011)] concerning subgraph component polynomials for cycles and complete graphs.

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Graphs & Digraphs

Cited by 458 publications

References 0 publications

Fixed point sets in digital topology, 2

Fixed point sets in digital topology, 2

Neighbor-locating colorings in graphs

On the weighted safe set problem on paths and cycles

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