Displaying similar documents to “Two classes of graphs related to extremal eccentricities”

Exact 2 -step domination in graphs

Gary Chartrand, Frank Harary, Moazzem Hossain, Kelly Schultz (1995)

Mathematica Bohemica

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For a vertex v in a graph G , the set N 2 ( v ) consists of those vertices of G whose distance from v is 2. If a graph G contains a set S of vertices such that the sets N 2 ( v ) , v ∈ S , form a partition of V ( G ) , then G is called a 2 -step domination graph. We describe 2 -step domination graphs possessing some prescribed property. In addition, all 2 -step domination paths and cycles are determined.

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...

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...

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...

Digraphs with isomorphic underlying and domination graphs: connected U G c ( d )

Kim A.S. Factor, Larry J. Langley (2007)

Discussiones Mathematicae Graph Theory

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The domination graph of a directed graph has an edge between vertices x and y provided either (x,z) or (y,z) is an arc for every vertex z distinct from x and y. We consider directed graphs D for which the domination graph of D is isomorphic to the underlying graph of D. We demonstrate that the complement of the underlying graph must have k connected components isomorphic to complete graphs, paths, or cycles. A complete characterization of directed graphs where k = 1 is presented. ...