20,012 research outputs found
On the metric dimension and fractional metric dimension for hierarchical product of graphs
A set of vertices {\em resolves} a graph if every vertex of is
uniquely determined by its vector of distances to the vertices in . The {\em
metric dimension} for , denoted by , is the minimum cardinality of
a resolving set of . In order to study the metric dimension for the
hierarchical product of two rooted graphs
and , we first introduce a new parameter, the {\em
rooted metric dimension} \rdim(G_1^{u_1}) for a rooted graph . If
is not a path with an end-vertex , we show that
\dim(G_2^{u_2}\sqcap G_1^{u_1})=|V(G_2)|\cdot\rdim(G_1^{u_1}), where
is the order of . If is a path with an end-vertex ,
we obtain some tight inequalities for .
Finally, we show that similar results hold for the fractional metric dimension.Comment: 11 page
Identifying codes of corona product graphs
For a vertex of a graph , let be the set of with all of
its neighbors in . A set of vertices is an {\em identifying code} of
if the sets are nonempty and distinct for all vertices . If
admits an identifying code, we say that is identifiable and denote by
the minimum cardinality of an identifying code of . In this
paper, we study the identifying code of the corona product of graphs
and . We first give a necessary and sufficient condition for the
identifiable corona product , and then express in terms of and the (total) domination number of .
Finally, we compute for some special graphs
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