A set $Wsubset V (G)$ is called a resolving set, if for every two distinct vertices $u, v in V (G)$ there exists $win W$ such that $d(u,w) not = d(v,w)$, where $d(x, y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$ with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish...

We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in AT-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in perm...

Given an undirected weighted graph G = (V,E) with vertex set V, edge set E and weights wi ∈ R associated to V or to E. Minimum weighted k-Cardinality tree problem (k-CARD for short) consists of finding a subtree of G with exactly k edges whose sum of weights is minimum [4]. There are two versions of this problem: vertex-weighted and edge-weighted, if weights to V or to E are associated, respect...

For a connected graph G = (V, E), a monophonic set of G is a set M � V (G) such that every vertex of G is contained in a monophonic path joining some pair of vertices in M. The monophonic number m (G) of G is the minimum order of its monophonic sets and any monophonic set of order m (G) is a minimum monophonic set of G. A connected monophonic set of a graph G is a monophonic set M such that the...

Let G = (V, E) be a connected undirected graph, where V = {1, . . . , n} is the set of nodes and E the set of edges. Consider that there are associated positive costs, ce, to each edge e ∈ E. Given a positive integer valued function d : V → N on the nodes, the Min-Degree Constrained Minimum Spanning Tree (md-MST) problem consists in finding a spanning tree T of G with minimum total edge cost, g...

In this paper, we investigate the structure of minimum vertex and edge cuts of distance-regular digraphs. We show that each distance-regular digraph Γ, different from an undirected cycle, is super edge-connected, that is, any minimum edge cut of Γ is the set of all edges going into (or coming out of) a single vertex. Moreover, we will show that except for undirected cycles, any distance regular...

‎‎Let $G=(V‎, ‎E)$ be a simple graph with vertex set $V$ and edge set $E$‎. ‎A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacent‎‎or incident to at least one element $yin Vcup E$ for which $f(y)=2$‎. ‎The weight of an‎‎MRDF $f$ is $sum _{xin Vcup E} f(x)$‎. ‎The mi...

A set D of vertices in a graph G is a distance-k dominating set if every vertex of G either is in D or is within distance k of at least one vertex in D. A distance-k dominating set of G of minimum cardinality is called a minimum distance-k dominating set of G. For any graph G and for a subset F of the edge set of G the set F is an edge dominating set of G if every edge of G either is in D or is...

Given a simple graph G = (V, E), an edge (u, u) E E is said to dominate itself and any edge (u,x) or (u,x), where x E V. A subset D C E is called an efficient edge dominating set of G if all edges in E are dominated by exactly one edge of D. The efficient edge domination problem is to find an efficient edge dominating set of minimum size in G. Suppose that each edge e E E is associated with a r...

Given a graph G, the Minimum Dominating Trail Set (MDTS) problem consists in ̄nding a minimum cardinality collection of pairwise edge-disjoint trails such that each edge of G has at least one endvertex on some trail. The MDTS problem is NP{hard for general graphs. In this paper lower and upper bounds for the MDTS problem on general graphs are presented.