Extremal mixed metric dimension with respect to the cyclomatic number
Introduction
Let be a simple connected graph with vertices and edges. The distance between a pair of vertices is defined as the length of the shortest path connecting and in and is denoted by . The distance between a vertex and an edge is defined by . For both these distances we simply write and if no confusion arises. We say that a vertex distinguishes (or resolves) a pair if We say that a set is a mixed metric generator if every pair is distinguished by at least one vertex from . The cardinality of the smallest mixed metric generator is called the mixed metric dimension of and it is denoted by
The notion of the mixed metric dimension is the natural generalization of the notions of the vertex metric dimension and the edge metric dimension which are defined as the cardinality of the smallest set of vertices which distinguishes all pairs of vertices and all pairs of edges respectively. The notion of vertex metric dimension for graphs was independently introduced by Harary and Melter [3] and [13], under the names resolving sets and locating sets, respectively. Even before, this notion was introduced for the realm of metric spaces [1]. The concept of vertex metric dimension [[2], [7], [10]] was recently extended from resolving vertices to resolving edges of a graph by Kelenc et al. [5], which lead to the definition of the edge metric dimension. Finally, it was further extended to resolving mixed pairs of edges and vertices by Kelenc et al. [6] which resulted with the notion of the mixed metric dimension. All these variations of metric dimensions attracted interest (see [[8], [9], [11], [12], [14], [15], [16]]), while for a wider and systematic introduction of the topic metric dimension that encapsulates all three above mentioned variations, we recommend the Ph.D. thesis of Kelenc [4].
In literature, among other questions, the mixed metric dimension of trees, unicyclic graphs and graphs with edge disjoint cycles was studied. Denoting by the number of leaves in a graph we first cite the following result from [6]. Proposition 1 For every tree it holds
A graph in which all cycles are pairwise edge disjoint is called a cactus graph. Having that in mind, the following results were proven in [12], first for unicyclic graphs and after for all cactus graphs. Theorem 2 Let be a cactus graph with cycles. Thenand the upper bound is attained if and only if every cycle in has exactly one vertex of degree .
The cyclomatic number of a graph is defined by As the number of cycles in trees and graphs with edge disjoint cycles equals the cyclomatic number, this lead the authors of [12] to make the following conjecture. Conjecture 3 Let be a graph, its cyclomatic number, and the number of leaves in . Then
Notice that Proposition 1 and Theorem 2 imply that the equality in (1) holds for all cactus graphs in which every cycle has precisely one vertex of degree (this includes all trees and unicyclic graphs with precisely one vertex on the cycle with degree ). A natural question that arises is - are there any other graphs for which the equality in (1) holds? In this paper we will try to further clarify this question.
Section snippets
Preliminaries
The (vertex) connectivity of a graph is the minimum size of a vertex cut, i.e. any subset of vertices such that is disconnected or has only one vertex. We say that a graph is -connected if As we are going to study the graphs for which the equality in (1) holds, it is useful to state the following result from [12]. Proposition 4 Let be a 3-connected graph. Then
This proposition implies that equality in (1) may hold only for graphs with (beside cactus graphs
Balanced -graphs
Notice that every -graph has the cyclomatic number and the number of leaves in such graph equals zero, i.e. Therefore, for a -graph the equality in (1) will hold if and only if In this section we will show that for balanced -graphs precisely this holds, i.e. if and only if is balanced. First we need the following lemma. Lemma 6 Let be a balanced -graph with vertices and of degree 3. Let be a set of vertices in such that and contains
Unbalanced -graphs
To complete the results we will now prove that Conjecture 3 holds also for unbalanced -graphs, but for them the equality in (1) does not hold. Lemma 9 Let be an unbalanced -graph, then Proof Let and be the two vertices of degree 3 in and let and be three distinct paths in connecting vertices and where without loss of generality we may assume that Since is an unbalanced -graph, it follows that By we denote the cycle induced by paths and
Concluding remarks
In [12] it was conjectured that for all graphs, where is the cyclomatic number and the number of leaves in a graph (see Conjecture 3). In this paper we focused our interest on graphs for which the conjecture holds with equality. It was already proven in literature that the equality holds for all trees, even more for all cactus graphs in which every cycle has precisely one vertex of degree . We wanted to find other graphs for which the equality holds. By
Acknowledgements
The authors acknowledge partial support Slovenian research agency ARRS program P1–0383 and ARRS project J1-1692 and also Project KK.01.1.1.02.0027, a project co-financed by the Croatian Government and the European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme.
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