4 research outputs found
Maximal proper subgraphs of median graphs
AbstractFor a median graph G and a vertex v of G that is not a cut-vertex we show that G-v is a median graph precisely when v is not the center of a bipartite wheel, which is in turn equivalent with the existence of a certain edge elimination scheme for edges incident with v. This implies a characterization of vertex-critical (respectively, vertex-complete) median graphs, which are median graphs whose all vertex-deleted subgraphs are not median (respectively, are median). Moreover, two analogous characterizations for edge-deleted median graphs are given
Blocks and Cut Vertices of the Buneman Graph
Given a set \Sg of bipartitions of some finite set of cardinality at
least 2, one can associate to \Sg a canonical -labeled graph \B(\Sg),
called the Buneman graph. This graph has several interesting mathematical
properties - for example, it is a median network and therefore an isometric
subgraph of a hypercube. It is commonly used as a tool in studies of DNA
sequences gathered from populations. In this paper, we present some results
concerning the {\em cut vertices} of \B(\Sg), i.e., vertices whose removal
disconnect the graph, as well as its {\em blocks} or 2-{\em connected
components} - results that yield, in particular, an intriguing generalization
of the well-known fact that \B(\Sg) is a tree if and only if any two splits
in \Sg are compatible
On a generalization of median graphs: -median graphs
Median graphs are connected graphs in which for all three vertices there is a
unique vertex that belongs to shortest paths between each pair of these three
vertices. To be more formal, a graph is a median graph if, for all , it holds that where
denotes the set of all vertices that lie on shortest paths connecting
and . In this paper we are interested in a natural generalization of
median graphs, called -median graphs. A graph is a -median graph, if
there are vertices such that, for all , it holds that , . By definition, every median graph with vertices is an -median graph.
We provide several characterizations of -median graphs that, in turn, are
used to provide many novel characterizations of median graphs