Minimum Weight Resolving Sets of Grid Graphs

For a simple graph $G=(V,E)$ and for a pair of vertices $u,v \in V$, we say that a vertex $w \in V$ resolves $u$ and $v$ if the shortest path from $w$ to $u$ is of a different length than the shortest path from $w$ to $v$. A set of vertices ${R \subseteq V}$ is a resolving set if for every pair of vertices $u$ and $v$ in $G$, there exists a vertex $w \in R$ that resolves $u$ and $v$. The minimum weight resolving set problem is to find a resolving set $M$ for a weighted graph $G$ such that$\sum_{v \in M} w(v)$ is minimum, where $w(v)$ is the weight of vertex $v$. In this paper, we explore the possible solutions of this problem for grid graphs $P_n \square P_m$ where $3\leq n \leq m$. We give a complete characterisation of solutions whose cardinalities are 2 or 3, and show that the maximum cardinality of a solution is $2n-2$. We also provide a characterisation of a class of minimals whose cardinalities range from $4$ to $2n-2$.Comment: 21 pages, 10 figure

An overview of the degree/diameter problem for directed, undirected and mixed graphs

A well-known fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, respectively, maximum outdegree, respectively, mixed degree; and given diameter. General upper bounds, called Moore bounds, exist for the largest possible order of such graphs, digraphs and mixed graphs of given maximum degree d (respectively, maximum out-degree d, respectively, maximum mixed degree) and diameter k. In recent years, there have been many interesting new results in all these three versions of the problem, resulting in improvements in both the lower bounds and the upper bounds on the largest possible number of vertices. However, quite a number of questions regarding the degree/diameter problem are still wide open. In this paper we present an overview of the current state of the degree/diameter problem, for undirected, directed and mixed graphs, and we outline several related open problems.Peer Reviewe

On The Existence of Non-Diregular Digraphs of Order Two Less than the Moore Bound

A communication network can be modelled as a graph or a directed graph, where each processing element is represented by a vertex and the connection between two processing elements is represented by an edge (or, in case of directed connections, by an arc). When designing a communication network, there are several criteria to be considered. For example, we can require an overall balance of the system. Given that all the processing elements have the same status, the flow of information and exchange of data between processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element, that is, if there is a balance (or regularity) in the network. This means that the in-degree and out-degree of each vertex in a directed graph (digraph) must be regular. In this paper, we present the existence of digraphs which are not diregular (regular out-degree, but not regular in-degree) with the number of vertices two less than the unobtainable upper bound for most values of out-degree and diameter, the so-called Moore bound

A family of mixed graphs with large order and diameter 2

A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order. We present a construction that provides mixed graphs of undirected degree qq, directed degree View the MathML sourceq-12 and order 2q22q2, for qq being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to View the MathML source9q2-4q+34 the defect of these mixed graphs is View the MathML source(q-22)2-14. In particular we obtain a known mixed Moore graph of order 1818, undirected degree 33 and directed degree 11 called Bosák’s graph and a new mixed graph of order 5050, undirected degree 55 and directed degree 22, which is proved to be optimal.Peer ReviewedPostprint (author's final draft

On large bipartite graphs of diameter 3

We consider the bipartite version of the {\it degree/diameter problem}, namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number $\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter $D$. In this context, the bipartite Moore bound \M^b(d,D) represents a general upper bound for $\N^b(d,D)$. Bipartite graphs of order \M^b(d,D) are very rare, and determining $\N^b(d,D)$ still remains an open problem for most $(d,D)$ pairs. This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on bipartite $(d,D,-4)$-graphs (that is, bipartite graphs of order \M^b(d,D)-4) was carried out. Here we first present some structural properties of bipartite $(d,3,-4)$-graphs, and later prove there are no bipartite $(7,3,-4)$-graphs. This result implies that the known bipartite $(7,3,-6)$-graph is optimal, and therefore $\N^b(7,3)=80$. Our approach also bears a proof of the uniqueness of the known bipartite $(5,3,-4)$-graph, and the non-existence of bipartite $(6,3,-4)$-graphs. In addition, we discover three new largest known bipartite (and also vertex-transitive) graphs of degree 11, diameter 3 and order 190, result which improves by 4 vertices the previous lower bound for $\N^b(11,3)$

Eccentric digraphs

AbstractThe distance d(u,v) from vertex u to vertex v in a digraph G is the length of the shortest directed path from u to v. The eccentricity e(v) of vertex v is the maximum distance of v to any other vertex of G. A vertex u is an eccentric vertex of vertex v if the distance from v to u is equal to the eccentricity of v. The eccentric digraph ED(G) of a digraph G is the digraph that has the same vertex set as G and the arc set defined by: there is an arc from u to v iff v is an eccentric vertex of u. The idea of the eccentric digraph of a graph was introduced by Buckley (Congr. Numer. 149 (2001) 65) and the idea of the eccentric digraph of a digraph by Boland and Miller (Proceedings of AWOCA’01, July 2001, p. 66). In this paper, we examine eccentric digraphs of digraphs for various families of digraphs and we consider the behaviour of an iterated sequence of eccentric digraphs of a digraph. The paper concludes with several open problems

On Total Vertex Irregularity Strength of Cocktail Party Graph

A vertex irregular total k-labeling of a graph G is a function λ from both the vertex and the edge sets to {1,2,3,,k} such that for every pair of distinct vertices u and x, λ(u)+∑λ(uv)≠λ(x)+∑λ(xy). uv∈E xy∈E. The integer k is called the total vertex irregularity strength, denoted by tvs (G ) , is the minimum value of the largest label over all such irregular assignments. In this paper, we prove that the total vertex irregularity strength of the Cocktail Party graph H2,n ,that is tvs(H2,n )= 3 for n ≥ 3

The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh

The problem of finding the largest connected subgraph of a given undirected host graph, subject to constraints on the maximum degree $\Delta$ and the diameter $D$, was introduced in \cite{maxddbs}, as a generalization of the Degree-Diameter Problem. A case of special interest is when the host graph is a common parallel architecture. Here we discuss the case when the host graph is a $k$-dimensional mesh. We provide some general bounds for the order of the largest subgraph in arbitrary dimension $k$, and for the particular cases of $k=3, \Delta = 4$ and $k=2, \Delta = 3$, we give constructions that result in sharper lower bounds.Comment: accepted, 18 pages, 7 figures; Discrete Applied Mathematics, 201

A characterization of strongly chordal graphs

AbstractIn this paper, we present a simple charactrization of strongly chordal graphs. A chordal graph is strongly chordal if and only if every cycle on six or more vertices has an induced triangle with exactly two edges of the triangle as the chords of the cycle