On the Metric Dimension of Cartesian Products of Graphs

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded

One size resolvability of graphs

For an ordered set W = {w1,w2, · · · ,wk} of vertices in a connected graph G and a vertex v of G, the code of v with respect to W is the k-vector CW(v) = (d(v,w1), d(v,w2), · · · , d(v,wk)). The set W is a one size resolving set for G if (1) the size of subgraph hWi induced by W is one and (2) distinct vertices of G have distinct code with respect to W. The minimum cardinality of a one size resolving set in graph G is the one size resolving number, denoted by or(G). A one size resolving set of cardinality or(G) is called an or-set of G. We study the existence of or-set in graphs and characterize all nontrivial connected graphs G of order n with or(G) = n and n − 1

The cardinality of endomorphisms of some oriented paths: an algorithm

An endomorphism of a (oriented) graph is a mapping on the vertex set preserving (arcs) edges. In this paper we provide an algorithm to determine the cardinalities of endomorphism monoids of some ( nite) directed paths, based on results on simple paths.Chiang Mai University; CMUC - Centro de Matemática da Universidade de Coimbra; Srinakharinwirot University; Thailand Research Fund and Commission on Higher Education, Thailand MRG508007

The independent resolving number of a graph

summary:For an ordered set $W=\lbrace w_1, w_2, \dots , w_k\rbrace$ of vertices in a connected graph $G$ and a vertex $v$ of $G$, the code of $v$ with respect to $W$ is the $k$-vector $$c_W(v) = (d(v, w_1), d(v, w_2), \dots , d(v, w_k) ).$$ The set $W$ is an independent resolving set for $G$ if (1) $W$ is independent in $G$ and (2) distinct vertices have distinct codes with respect to $W$. The cardinality of a minimum independent resolving set in $G$ is the independent resolving number $\mathop {\mathrm ir}(G)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order $n$ with $\mathop {\mathrm ir}(G) = 1$, $n-1$, $n-2$, and present several realization results. It is shown that for every pair $r, k$ of integers with $k \ge 2$ and $0 \le r \le k$, there exists a connected graph $G$ with $\mathop {\mathrm ir}(G) = k$ such that exactly $r$ vertices belong to every minimum independent resolving set of $G$

Network discovery and verification

Abstract. Consider the problem of discovering (or verifying) the edges and non-edges of a network, modelled as a connected undirected graph, using a minimum number of queries. A query at a vertex v discovers (or verifies) all edges and non-edges whose endpoints have different distance from v. In the network discovery problem, the edges and non-edges are initially unknown, and the algorithm must select the next query based only on the results of previous queries. We study the problem using competitive analysis and give a randomized on-line algorithm with competitive ratio O ( √ n log n) for graphs with n vertices. We also show that no deterministic algorithm can have competitive ratio better than 3. In the network verification problem, the graph is known in advance and the goal is to compute a minimum number of queries that verify all edges and nonedges. This problem has previously been studied as the problem of placing landmarks in graphs or determining the metric dimension of a graph. We show that there is no approximation algorithm for this problem with ratio o(log n) unless P = N P. Furthermore, we prove that the optimal number of queries for d-dimensional hypercubes is Θ(d / log d).