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 ...

An H-decomposition of a graph G is a partition of the edge-set of G into subsets, where each subset induces a copy of the graph H. 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 distinct H-decompositions intersect in at most one edge. In case G=Kn and H=Kr , a k-orthogonal Kr -decomposition of Kn is called an (n, r, k) completel...

For an ordered k-decomposition D = {G1, G2,...,Gk} of a connected graph G (V,E), the D-representation edge e is k-tuple γ(e/D)=(d(e, G1), d(e, G2), ...,d(e, Gk)), where Gi) represents distance from to Gi. A decomposition resolving if every two distinct edges have representations. The minimum k for which has its dimension dec(G). In this paper, corona product path Pn and cycle Cn with complete g...

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.

In this paper we study monophonic sets in a connected graph G. First, we present a realization theorem proving, that there is no general relationship between monophonic and geodetic hull sets. Second, we study the contour of a graph, introduced by Cáceres and alt. [2] as a generalization of the set of extreme vertices where the authors proved that the contour of a graph is a g-hull set; in this...

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 graceful labeling of a graph $G=(V,E)$ with $m$ edges is aninjection $f: V(G) rightarrow {0,1,ldots,m}$ such that the resulting edge labelsobtained by $|f(u)-f(v)|$ on every edge $uv$ are pairwise distinct. For natural numbers $n$ and $k$, where $n > 2k$, a generalized Petersengraph $P(n, k)$ is the graph whose vertex set is ${u_1, u_2, cdots, u_n} cup {v_1, v_2, cdots, v_n}$ and its edge set...

Let G be a subgraph of Kn. The graph obtained from G by replacing each edge with a 3-cycle whose third vertex is distinct from other vertices in the configuration is called a T (G)-triple. An edge-disjoint decomposition of 3Kn into copies of T (G) is called a T (G)-triple system of order n. If, in each copy of T (G) in a T (G)triple system, one edge is taken from each 3-cycle (chosen so that th...