Let G be an undirected graph with vertex and edge sets V (G) E(G), respectively. A subset S of vertices is a geodetic hop dominating set if it both set. The domination number G, γhg(G), the minimum cardinality among all in G. Geodetic resulting from some binary operations have been characterized. These characterizations used to determine tight bounds for each graphs considered.

A vertex set D in graph G is called a geodetic set if all vertices of G are lying on some shortest u–v path of G, where u, v 2 D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing ...

A cube decomposition Q of a graph G is said to be 2-perfect if for every edge {x, y} ∈ E(G), x and y are connected by a path of length 1 in exactly one cube of Q, and are also connected by a path of length 2 in exactly one (distinct) cube of Q. For both Kv and Kv −F , we give constructions for half of the cases which satisfy the obvious necessary conditions, with a small number of exceptions. W...

We present an algorithm that on input of an n×n symmetric diagonally dominant matrix A with m non-zero entries constructs in time Õ(m log n) a solver which on input of a vector b computes a vector x satisfying ||x−Ab||A < �||Ab||A in time Õ(m log n log(1/�)) 1. The new algorithm exploits previously unknown structural properties of the output of the incremental sparsification algorithm given in ...

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in at least one interval between the vertices of S. The size of a minimum geodetic set in G is the geodetic number of G. Upper bounds for the geodetic number of Cartesian product graphs are proved and for several classes exact values are obtained. It is proved that many metrically defined sets in Cartesian products hav...

An H-decomposition of a graph G is a set L of edge-disjoint Hsubgraphs of G, such that each edge of G appears in some element of L. A k-orthogonal H-decomposition of a graph G is a set of k H-decompositions of G, such that any two copies of H in any two distinct H-decompositions have at most one edge in common. We prove that for every fixed graph H and every fixed integer k ≥ 1, if n is suffici...

Let R be a commutative ring with $Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $T(Gamma(R))$. It is the (undirected) graph with all elements of R as vertices, and for distinct $x, yin R$, the vertices $x$ and $y$ are adjacent if and only if $x + yinZ(R)$. We study the chromatic number and edge connectivity of this graph.

Introducing certain types of morphisms for general (abstract) convexity spaces, we give several ways for reducing the (Calder-) Eckhoff partition problem to simpler equivalent forms (finite, point-convex, interval spaces; restricted form, i.e. with distinct points). With additional results (to appear in a forthcoming paper) we show how the general problem can be reduced to its restricted versio...

For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u − v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u, v) for u, v ∈ S. A set S is a geodetic set if I(S) = V (G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set...

A subset S of vertices in a graph G is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A subset D of vertices in G is called dominating set if every vertex not in D has at least one neighbor in D. A geodetic dominating set S is both a geodetic and a dominating set. The geodetic (domination, geodetic domination) number g(G)(γ(G), γg(G)) of G is...