23,955 research outputs found
Resolving sets for breaking symmetries of graphs
This paper deals with the maximum value of the difference between the
determining number and the metric dimension of a graph as a function of its
order. Our technique requires to use locating-dominating sets, and perform an
independent study on other functions related to these sets. Thus, we obtain
lower and upper bounds on all these functions by means of very diverse tools.
Among them are some adequate constructions of graphs, a variant of a classical
result in graph domination and a polynomial time algorithm that produces both
distinguishing sets and determining sets. Further, we consider specific
families of graphs where the restrictions of these functions can be computed.
To this end, we utilize two well-known objects in graph theory: -dominating
sets and matchings.Comment: 24 pages, 12 figure
Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs
A total dominating set in a graph is a set of vertices such that every vertex
of the graph has a neighbor in the set. We introduce and study graphs that
admit non-negative real weights associated to their vertices such that a set of
vertices is a total dominating set if and only if the sum of the corresponding
weights exceeds a certain threshold. We show that these graphs, which we call
total domishold graphs, form a non-hereditary class of graphs properly
containing the classes of threshold graphs and the complements of domishold
graphs, and are closely related to threshold Boolean functions and threshold
hypergraphs. We present a polynomial time recognition algorithm of total
domishold graphs, and characterize graphs in which the above property holds in
a hereditary sense. Our characterization is obtained by studying a new family
of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of
independent interest.Comment: 19 pages, 1 figur
Maximum matching width: new characterizations and a fast algorithm for dominating set
We give alternative definitions for maximum matching width, e.g. a graph
has if and only if it is a subgraph of a chordal
graph and for every maximal clique of there exists with and such that any subset of
that is a minimal separator of is a subset of either or .
Treewidth and branchwidth have alternative definitions through intersections of
subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We
show that mm-width combines both aspects, focusing on nodes and on edges. Based
on this we prove that given a graph and a branch decomposition of mm-width
we can solve Dominating Set in time , thereby beating
whenever . Note that and these inequalities are
tight. Given only the graph and using the best known algorithms to find
decompositions, maximum matching width will be better for solving Dominating
Set whenever
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