54 research outputs found
Locating-dominating sets and identifying codes in graphs of girth at least 5
Locating-dominating sets and identifying codes are two closely related
notions in the area of separating systems. Roughly speaking, they consist in a
dominating set of a graph such that every vertex is uniquely identified by its
neighbourhood within the dominating set. In this paper, we study the size of a
smallest locating-dominating set or identifying code for graphs of girth at
least 5 and of given minimum degree. We use the technique of vertex-disjoint
paths to provide upper bounds on the minimum size of such sets, and construct
graphs who come close to meet these bounds.Comment: 20 pages, 9 figure
Partitions of graphs into small and large sets
Let be a graph on vertices. We call a subset of the vertex set
\emph{-small} if, for every vertex , . A subset is called \emph{-large} if, for every vertex
, . Moreover, we denote by the
minimum integer such that there is a partition of into -small
sets, and by the minimum integer such that there is a
partition of into -large sets. In this paper, we will show tight
connections between -small sets, respectively -large sets, and the
-independence number, the clique number and the chromatic number of a graph.
We shall develop greedy algorithms to compute in linear time both
and and prove various sharp inequalities
concerning these parameters, which we will use to obtain refinements of the
Caro-Wei Theorem, the Tur\'an Theorem and the Hansen-Zheng Theorem among other
things.Comment: 21 page
New approach to the k-independence number of a graph
Let G = (V,E) be a graph and k > 0 an integer. A k-independent set S V is a set of vertices such that the maximum degree in the graph induced by S is at most k. With k(G) we denote the maximum cardinality of a k-independent set of G. We prove that, for a graph G on n vertices and average degree d, k(G) > k+1 dde+k+1n, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99–107].Peer ReviewedPostprint (published version
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