8 research outputs found
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
The generalized 3-edge-connectivity of lexicographic product graphs
The generalized -edge-connectivity of a graph is a
generalization of the concept of edge-connectivity. The lexicographic product
of two graphs and , denoted by , is an important graph
product. In this paper, we mainly study the generalized 3-edge-connectivity of
, and get upper and lower bounds of .
Moreover, all bounds are sharp.Comment: 14 page
Generalized (edge-)connectivity of join, corona and cluster graphs
The generalized -connectivity of a graph , introduced by Hager in 1985, is a natural generalization of the classical connectivity. As a natural counterpart, Li et al. proposed the concept of generalized -edge-connectivity . In this paper, we obtain exact values or sharp upper and lower bounds of and for join, corona and cluster graphs
Identifying Codes and Domination in the Product of Graphs
An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code in a graph is denoted \gid(G). We consider identifying codes of the direct product . In particular, we answer a question of Klav\v{z}ar and show the exact value of \gid(K_n \times K_m). It was recently shown by Gravier, Moncel and Semri that for the Cartesian product of two same-sized cliques \gid(K_n \Box K_n) = \lfloor{\frac{3n}{2}\rfloor}. Letting be any integers, we show that \IDCODE(K_n \cartprod K_m) = \max\{2m-n, m + \lfloor n/2 \rfloor\}. Furthermore, we improve upon the bounds for \IDCODE(G \cartprod K_m) and explore the specific case when is the Cartesian product of multiple cliques. Given two disjoint copies of a graph , denoted and , and a permutation of , the permutation graph is constructed by joining to for all . The graph is said to be a universal fixer if the domination number of is equal to the domination number of for all of . In 1999 it was conjectured that the only universal fixers are the edgeless graphs. We prove the conjecture