6 research outputs found
On The Center Sets and Center Numbers of Some Graph Classes
For a set of vertices and the vertex in a connected graph ,
is called the -eccentricity of in
. The set of vertices with minimum -eccentricity is called the -center
of . Any set of vertices of such that is an -center for some
set of vertices of is called a center set. We identify the center sets
of certain classes of graphs namely, Block graphs, , , wheel
graphs, odd cycles and symmetric even graphs and enumerate them for many of
these graph classes. We also introduce the concept of center number which is
defined as the number of distinct center sets of a graph and determine the
center number of some graph classes
The median function on graphs with bounded profiles
AbstractThe median of a profile π=(u1,…,uk) of vertices of a graph G is the set of vertices x that minimize the sum of distances from x to the vertices of π. It is shown that for profiles π with diameter θ the median set can be computed within an isometric subgraph of G that contains a vertex x of π and the r-ball around x, where r>2θ−1−2θ/|π|. The median index of a graph and r-joins of graphs are introduced and it is shown that r-joins preserve the property of having a large median index. Consensus strategies are also briefly discussed on a graph with bounded profiles
Fair Sets of Some Class of Graphs
Given a non empty set of vertices of a graph, the partiality of a vertex
with respect to is the difference between maximum and minimum of the
distances of the vertex to the vertices of . The vertices with minimum
partiality constitute the fair center of the set. Any vertex set which is the
fair center of some set of vertices is called a fair set. In this paper we
prove that the induced subgraph of any fair set is connected in the case of
trees and characterise block graphs as the class of chordal graphs for which
the induced subgraph of all fair sets are connected. The fair sets of ,
, , wheel graphs, odd cycles and symmetric even graphs are
identified. The fair sets of the Cartesian product graphs are also discussed.Comment: 14 pages, 4 figure
Centers and medians of distance-hereditary graphs
AbstractA graph is distance-hereditary if the distance between any two vertices in a connected induced subgraph is the same as in the original graph. In this paper, we study metric properties of distance-hereditary graphs. In particular, we determine the structures of centers and medians of distance-hereditary and related graphs. The relations between eccentricity, radius, and diameter of such graphs are also investigated
Centers and medians of distance-hereditary graphs
A graph is distance-hereditary ifthe distance between any two vertices in a connected induced
subgraph is the same as in the original graph. In this paper, we study metric properties of
distance-hereditary graphs. In particular, we determine the structures ofcenters & medians of
distance-hereditary & related graphs. The relations between eccentricity, radius, & diameter
ofsuch graphs are also investigated