Extremal mixed metric dimension with respect to the cyclomatic number

doi:10.1016/j.amc.2021.126238

Applied Mathematics and Computation

Volume 404, 1 September 2021, 126238

https://doi.org/10.1016/j.amc.2021.126238 Get rights and content

Highlights

•
Prove that for unbalanced $Θ$ -graphs the strict inequality holds.
•
Prove that for balanced $Θ$ -graphs the equality is attained.
•
Further conjecture that balanced $Θ$ -graphs, besides trees and cactus graph in which every cycle has precisely one vertex of degree $\geq$ 3 for which it was already known that equality holds, are the only graphs for which this upper bound is attained.

Abstract

In a graph $G,$ the cardinality of the smallest ordered set of vertices that distinguishes every element of $V (G) \cup E (G)$ is called the mixed metric dimension of $G,$ and it is denoted by $mdim (G) .$ In [12] it was conjectured that every graph $G$ with cyclomatic number $c (G)$ satisfies $mdim (G) \leq L_{1} (G) + 2 c (G)$ where $L_{1} (G)$ is the number of leaves in $G$ . It is already proven that the equality holds for all trees and more generally for graphs with edge-disjoint cycles in which every cycle has precisely one vertex of degree $\geq 3$ . In this paper we determine that for every $Θ$ -graph $G,$ the mixed metric dimension $mdim (G)$ equals 3 or 4, with 4 being attained if and only if $G$ is a balanced $Θ$ -graph. Thus, for balanced $Θ$ -graphs the above inequality is also tight. We conclude the paper by further conjecturing that there are no other graphs, besides the ones mentioned here, for which the equality $mdim (G) = L_{1} (G) + 2 c (G)$ holds.

Introduction

Let $G$ be a simple connected graph with $n$ vertices and $m$ edges. The distance between a pair of vertices $u, v \in V (G)$ is defined as the length of the shortest path connecting $u$ and $v$ in $G$ and is denoted by $d_{G} (u, v)$ . The distance between a vertex $u \in V (G)$ and an edge $e = v w \in E (G)$ is defined by $d_{G} (u, e) = d_{G} (u, v w) = \min {d_{G} (u, v), d_{G} (u, w)}$ . For both these distances we simply write $d (u, v)$ and $d (u, e)$ if no confusion arises. We say that a vertex $s \in V (G)$ distinguishes (or resolves) a pair $x, x^{'} \in V (G) \cup E (G)$ if $d (s, x) \neq d (s, x^{'}) .$ We say that a set $S \subseteq V (G)$ is a mixed metric generator if every pair $x, x^{'} \in V (G) \cup E (G)$ is distinguished by at least one vertex from $S$ . The cardinality of the smallest mixed metric generator is called the mixed metric dimension of $G,$ and it is denoted by $mdim (G) .$

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 $L_{1} (G)$ the number of leaves in a graph $G,$ we first cite the following result from [6].

Proposition 1

For every tree $T,$ it holds $mdim (T) = L_{1} (T) .$

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 $G \neq C_{n}$ be a cactus graph with $c$ cycles. Then $mdim (G) \leq L_{1} (G) + 2 c,$ and the upper bound is attained if and only if every cycle in $G$ has exactly one vertex of degree $\geq 3$ .

The cyclomatic number of a graph $G$ is defined by $c (G) = m - n + 1 .$ 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 $G \neq C_{n}$ be a graph, $c (G)$ its cyclomatic number, and $L_{1} (G)$ the number of leaves in $G$ . Then $mdim (G) \leq L_{1} (G) + 2 c (G) .$

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 $\geq 3$ (this includes all trees and unicyclic graphs with precisely one vertex on the cycle with degree $\geq 3$ ). 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 $κ (G)$ of a graph $G$ is the minimum size of a vertex cut, i.e. any subset of vertices $S \subseteq V (G)$ such that $G - S$ is disconnected or has only one vertex. We say that a graph $G$ is $k$ -connected if $κ (G) \geq k .$ 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 $G$ be a 3-connected graph. Then $mdim (G) < 2 c (G) .$

This proposition implies that equality in (1) may hold only for graphs with $κ (G) = 1$ (beside cactus graphs

Balanced $Θ$ -graphs

Notice that every $Θ$ -graph $G$ has the cyclomatic number $c (G) = 2$ and the number of leaves in such graph equals zero, i.e. $L_{1} (G) = 0 .$ Therefore, for a $Θ$ -graph $G,$ the equality in (1) will hold if and only if $mdim (G) = 4 .$ In this section we will show that for balanced $Θ$ -graphs precisely this holds, i.e. $mdim (G) = 4$ if and only if $G$ is balanced. First we need the following lemma.

Lemma 6

Let $G$ be a balanced $Θ$ -graph with vertices $u$ and $v$ of degree 3. Let $S \subseteq V (G)$ be a set of vertices in $G$ such that $| S | = 3$ and $S$ contains

Unbalanced $Θ$ -graphs

To complete the results we will now prove that Conjecture 3 holds also for unbalanced $Θ$ -graphs, but for them the equality in (1) does not hold.

Lemma 9

Let $G$ be an unbalanced $Θ$ -graph, then $mdim (G) \leq 3 .$

Proof

Let $u$ and $v$ be the two vertices of degree 3 in $G$ and let $P_{1},$ $P_{2}$ and $P_{3}$ be three distinct paths in $G$ connecting vertices $u$ and $v,$ where without loss of generality we may assume that $| P_{1} | \leq | P_{2} | \leq | P_{3} | .$ Since $G$ is an unbalanced $Θ$ -graph, it follows that $| P_{3} | - | P_{1} | \geq 2 .$ By $C_{i j}$ we denote the cycle induced by paths $P_{i}$ and

Concluding remarks

In [12] it was conjectured that $mdim (G) \leq L_{1} (G) + 2 c (G)$ for all graphs, where $c (G)$ is the cyclomatic number and $L_{1} (G)$ the number of leaves in a graph $G$ (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 $\geq 3$ . 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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Extremal mixed metric dimension with respect to the cyclomatic number

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Balanced Θ-graphs

Unbalanced Θ-graphs

Concluding remarks

Acknowledgements

Discret. Appl. Math.

Appl. Math. Comput.

Discret. Appl. Math.

Appl. Math. Comput.

Appl. Math. Comput.

Discret. Appl. Math.

Discret. Math.

Theory and applications of Distance Geometry

Balanced $Θ$ -graphs

Unbalanced $Θ$ -graphs