On the ensemble of optimal dominating and locating-dominating codes in a graph
Introduction
We introduce basic definitions and notation for graphs (for which we refer to, e.g., [2] and [4]), and for codes. Dominating codes constitute an old, large, classical topic (see, e.g., [5] or [6]); in the particular case when the graph is the hypercube, they are known as covering codes and have received a lot of attention in Coding Theory: see [3] and the on-line bibliography at [9], with 1000 references. Locating-dominating codes [12] are part of a larger class of codes which aim at distinguishing, in some ways, between vertices: watching systems, identifying, locating-dominating and discriminating codes, resolving sets, …; they may have many applications and are a fast growing field, as show the 300 references in the on-line bibliography at [10], most of them published in the 21st century.
We denote by a simple, undirected graph with vertex set V and edge set E, where an edge between and is denoted by xy or yx. Two vertices linked by an edge are said to be neighbours. We denote by the set of neighbours of the vertex v, and . An induced subgraph of G is a graph with vertex set and edge set . We say that two graphs and are isomorphic, and write , if there is a bijection such that if, and only if, for all .
Whenever three vertices are such that and , we say that z separates x and y in G (note that is possible). A set is said to separate x and y in G if it contains at least one vertex which does.
A code C is simply a subset of V, and its elements are called codewords. For each vertex , the identifying set of v, with respect to C, is denoted by and is defined by We say that C is a dominating code in G if all the sets , , are nonempty.
We say that C is a locating-dominating code [12] if all the sets , , are nonempty and distinct. In particular, any two non-codewords are separated by C. In the sequel, we shall use LD for locating-dominating.
We denote by (respectively, ) the smallest cardinality of a dominating (respectively, LD) code. Any dominating (respectively, LD) code C such that (respectively, ) is said to be optimal.
One application of LD codes is, for instance, fault diagnosis in multiprocessor systems: such a system can be modeled by a graph where V is the set of processors and E the set of links between processors. Assume that at most one processor is malfunctioning and we wish to test the system and locate the faulty processor. For this purpose, some processors (constituting the code) will be chosen and assigned the task of testing their neighbours. Whenever a selected processor (or codeword) detects a fault, it sends an alarm signal. We require that we can uniquely tell the location of the malfunctioning processor based on the information which ones of the codewords gave the alarm; under the assumption that the codewords work without failure, or that their only task is to test their neighbours (i.e., they are not considered as processors anymore) and that they perform this simple task without failure, then an LD code is what we need, because no two non-codewords have the same (nonempty) set of neighbours-codewords.
In this paper, we study the structure of the ensemble of all the optimal dominating codes and the ensemble of all the optimal LD codes of a graph. These ensembles are trivially collections of k-element subsets, or k-subsets, of V, for or ; we denote these ensembles by and , respectively. Conversely, assume that is a nonempty collection of some s different k-subsets of . The question is: is there a graph G with vertex set such that is equal to or ? When , the answer for almost all collections is NO; indeed, there are such collections but only different graphs. However, we can ask the same question for a graph G with vertices, . And now the answer is YES: Theorem 2 below states that
given any collection of k-subsets of , there is a positive integer m and a graph with , where , such that is an optimal dominating code in G if, and only if, for some .
Theorem 3 gives a similar result for LD codes, with similar consequences for ; also, the same kind of result is proved for identifying codes, which we do not define here, in [7].
Now, this establishes a sufficiently strong link, between the ensembles of the optimal dominating or LD codes of all graphs and the sets of k-subsets of n-sets, to connect our investigation to the following definition from [11] and the results related to it; see also [1]. Definition 1 Given positive integers k and n with , the Johnson graph is the graph whose vertex set consists of all the k-subsets of , with edges between two vertices sharing exactly elements. A graph H is isomorphic to an induced subgraph of a Johnson graph if, and only if, it is possible to assign, for some k and n, a k-subset to each vertex v of H in such a way that distinct vertices have distinct corresponding k-sets, and vertices v and w are neighbours if, and only if, and share exactly elements. In this case, we say that H is an induced subgraph of a Johnson graph, or that H is a JIS for short. We denote by the set of all induced subgraphs of all Johnson graphs.
If we link two elements and in (respectively, ) if, and only if, , then we obtain a graph which we denote by (respectively, ), and the set of all the graphs (respectively, ) is denoted by (respectively, ). Now, what Theorem 2, Theorem 3 show as an immediate consequence is that
every JIS belongs to , or: ;
every JIS belongs to , or: .
Section snippets
Main results
Theorem 2 Let be an arbitrary integer, and assume that is any nonempty collection of k-subsets of . Then there is a positive integer m and a graph G with vertex set , where , such that is an optimal dominating code in G if, and only if, for some .
Proof Denote by the set of all -subsets of together with all the k-subsets of that do not belong to ; this set has size . We begin the construction of G by taking n vertices
Some results on Johnson induced subgraphs
Some families of graphs are known to be JIS, some are known which are not JIS, but no characterization is available. Below, we summarize some of the results from [11]; for Cartesian products (see (d) below), we refer to [8].
Theorem 4 [Prop. 4] All complete graphs and all cycles are JIS; All trees are JIS; [Prop. 6] A graph is a JIS if, and only if, all its connected components are JIS; [Prop. 7] The Cartesian product of two JIS is a JIS; [Prop. 12] Any graph obtained by removing one edge from the complete
References (12)
Johnson graphs are Hamiltonian-connected
Ars Math. Contemp.
(2013)Graphes
(1983)(1985)- et al.
Covering Codes
(1997) Graph Theory
(2005)- et al.
Fundamentals of Domination in Graphs
(1998) - et al.
Total Domination in Graphs
(2013)
Cited by (8)
Unique (optimal) solutions: Complexity results for identifying and locating–dominating codes
2019, Theoretical Computer ScienceCitation Excerpt :In a forthcoming work, we extend the present study on uniqueness issues to Boolean satisfiability and graph colouring [16], Vertex Cover and Dominating Set (as well as its generalization to domination within distance r) [17], and Hamiltonian Cycle [18]. At the other end, there has been research on how many optimal r-identifying codes can exist in a graph [19], and on the structure of the ensemble of optimal r-locating–dominating codes [20] and of optimal r-identifying codes [21]. For other works in the area of complexity, see, e.g., [22], [23], [24], [25], [26], [27], and [28], which establish, in particular, polynomiality or NP-completeness results for the identification problem when restricted to some subclasses of graphs, such as trees, planar graphs, bipartite graphs, interval graphs or line graphs.
ON THE BINARY LOCATING-DOMINATION NUMBER OF REGULAR AND STRONGLY-REGULAR GRAPHS
2023, Journal of Mathematical InequalitiesWhich Graphs Occur as γ -Graphs?
2020, Graphs and CombinatoricsWhich graphs occur as γ-graphs?
2018, arXivThe binary locating-dominating number of some convex polytopes
2017, Ars Mathematica Contemporanea