Locating-dominating sets in hypergraphs

Abstract

A hypergraph is a generalization of a graph where edges can connect any number of vertices. In this paper, we extend the study of locating-dominating sets to hypergraphs. Along with some basic results, sharp bounds for the locationdomination number of hypergraphs in general and exact values with specified conditions are investigated. Moreover, locating-dominating sets in some specific hypergraphs are found. -complete problem [8]. In [6], it was pointed out that each locating-dominating set is both the locating and dominating set. However, a set that is both the locating and dominating is not necessarily a locating-dominating set. NPA hypergraph H is a pair (V (H), E(H)), where V (H) is a finite set of vertices and E(H) is a finite family of non-empty subsets of V (H), called hyperedges, withso in a linear hypergraph, there may be no repeated hyperedges of cardinality greater than one. A hypergraph H with no hyperedge is a subset of any other hyperedge is called Sperner.AIf v is incident with exactly n hyperedges, then we say that the degree of v is n; if all the vertices v ∈ V (H) have degree n, then H is n-regular. The maximum degree of any vertex in H is denoted by ∆(H). Similarly, if there are exactly n vertices incident with a hyperedge E, then we say that the size of E is n; if all the hyperedges E ∈ E(H) have size n, then H is n-uniform. A simple graph is a 2-uniform hypergraph.A path from a vertex v to another vertex u, in a hypergraph, is a finite sequence of the form v, , having length l such that v ∈ E 1 , w i ∈ E i ∩ E i+1 for i = 1, 2, . . . , l − 1 and u ∈ E l . A hypergraph H is connected if there is a path between every two vertices of H. All the hypergraphs considered in this paper are connected Sperner hypergraphs.A hypergraph H is said to be a hyperstar if E i ∩E j = C = ∅, for any E i , E j ∈ E(H). We will call C, the center of the hyperstar. If there exists a sequence of hyperedges E 1 , E 2 , . . . , E k in a hypergraph H, then H is said to be (1) a hyperpath if E i ∩E j = ∅ if and only if |i − j| = 1; (2) a hypercycle if E i ∩ E j = ∅ if and only if |i − j| = 1 (mod k). A connected hypergraph H with no hypercycle is called a hypertree.In graphs, the theory of dominating sets and locating-dominating sets is extensively studied. Hypergraphs, in the context of domination, were firstly considered by Behr and Camarinopoulos in 1998 [4], and further considered by Acharya [1,2] and Jose and Tuza [13]. In this paper, we consider hypergraphs in the context of location-domination. We give some sharp lower bounds for the location-domination number of hypergraphs. Also, we investigate the location-domination number of some well-known families of hypergraphs such as hyperpaths, hypercycles and kpartite hypergraphs. Some Basic Results and BoundsTwo vertices u and v of a hypergraph incident with the same hyperedge are said to be coincident vertices. Let E (d) i = {E i 1 , E i 2 , . . . , E i d } be a collection of hyperedges.

On Locating-Dominating Set of Regular Graphs

“…Charon et al [3] have proved that determining locatingdominating number of a graph is NP-complete problem which is reduced from 3-SAT. However, some results for certain classes of graphs have been obtained, such as paths [2], cycles [4], stars [5], complete graphs [6], bipartite graphs [7,8], complete multipartite graphs [5], wheels [7], twin-free graphs [9,10], and hypergraphs [11]. In [12], Balbuena et al investigated a locating-dominating set of graphs with girth at least 5.…”

On some variations of dominating identification in graphs

“…And classify all the graphs satisfying the upper bound. Fazil et al (2016) worked on the locationg-dominating sets in hyper graph.…”