Displaying similar documents to “Closed k-stop distance in graphs”

Radio number for some thorn graphs

Ruxandra Marinescu-Ghemeci (2010)

Discussiones Mathematicae Graph Theory

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For a graph G and any two vertices u and v in G, let d(u,v) denote the distance between u and v and let diam(G) be the diameter of G. A multilevel distance labeling (or radio labeling) for G is a function f that assigns to each vertex of G a positive integer such that for any two distinct vertices u and v, d(u,v) + |f(u) - f(v)| ≥ diam(G) + 1. The largest integer in the range of f is called the span of f and is denoted span(f). The radio number of G, denoted rn(G), is the minimum span...

Cores and shells of graphs

Allan Bickle (2013)

Mathematica Bohemica

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The k -core of a graph G , C k ( G ) , is the maximal induced subgraph H G such that δ ( G ) k , if it exists. For k > 0 , the k -shell of a graph G is the subgraph of G induced by the edges contained in the k -core and not contained in the ( k + 1 ) -core. The core number of a vertex is the largest value for k such that v C k ( G ) , and the maximum core number of a graph, C ^ ( G ) , is the maximum of the core numbers of the vertices of G . A graph G is k -monocore if C ^ ( G ) = δ ( G ) = k . This paper discusses some basic results on the structure of k -cores and...

On integral sum graphs with a saturated vertex

Zhibo Chen (2010)

Czechoslovak Mathematical Journal

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As introduced by F. Harary in 1994, a graph G is said to be an i n t e g r a l s u m g r a p h if its vertices can be given a labeling f with distinct integers so that for any two distinct vertices u and v of G , u v is an edge of G if and only if f ( u ) + f ( v ) = f ( w ) for some vertex w in G . We prove that every integral sum graph with a saturated vertex, except the complete graph K 3 , has edge-chromatic number equal to its maximum degree. (A vertex of a graph G is said to be if it is adjacent to every...

Minimum degree, leaf number and traceability

Simon Mukwembi (2013)

Czechoslovak Mathematical Journal

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Let G be a finite connected graph with minimum degree δ . The leaf number L ( G ) of G is defined as the maximum number of leaf vertices contained in a spanning tree of G . We prove that if δ 1 2 ( L ( G ) + 1 ) , then G is 2-connected. Further, we deduce, for graphs of girth greater than 4, that if δ 1 2 ( L ( G ) + 1 ) , then G contains a spanning path. This provides a partial solution to a conjecture of the computer program Graffiti.pc [DeLaVi na and Waller, Spanning trees with many leaves and average distance, Electron. J. Combin....

The independent resolving number of a graph

Gary Chartrand, Varaporn Saenpholphat, Ping Zhang (2003)

Mathematica Bohemica

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For an ordered set W = { w 1 , w 2 , , w k } 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 ) , , 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 i r ( G ) . We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs G of order...

On well-covered graphs of odd girth 7 or greater

Bert Randerath, Preben Dahl Vestergaard (2002)

Discussiones Mathematicae Graph Theory

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A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth...

Graphs with the same peripheral and center eccentric vertices

Peter Kyš (2000)

Mathematica Bohemica

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The eccentricity e ( v ) of a vertex v is the distance from v to a vertex farthest from v , and u is an eccentric vertex for v if its distance from v is d ( u , v ) = e ( v ) . A vertex of maximum eccentricity in a graph G is called peripheral, and the set of all such vertices is the peripherian, denoted P e r i G ) . We use C e p ( G ) to denote the set of eccentric vertices of vertices in C ( G ) . A graph G is called an S-graph if C e p ( G ) = P e r i ( G ) . In this paper we characterize S-graphs with diameters less or equal to four, give some constructions of...