2,911 research outputs found
In the complement of a dominating set
A set D of vertices of a graph G=(V,E) is a dominating set, if every vertex
of D\V has at least one neighbor that belongs to D. The disjoint domination
number of a graph G is the minimum cardinality of two disjoint dominating
sets of G. We prove upper bounds for the disjoint domination number for
graphs of minimum degree at least 2, for graphs of large minimum degree and
for cubic graphs.A set T of vertices of a graph G=(V,E) is a total
dominating set, if every vertex of G has at least one neighbor that belongs
to T. We characterize graphs of minimum degree 2 without induced 5-cycles
and graphs of minimum degree at least 3 that have a dominating set, a total
dominating set, and a non-empty vertex set that are disjoint.A set I of
vertices of a graph G=(V,E) is an independent set, if all vertices in I are
not adjacent in G. We give a constructive characterization of trees that
have a maximum independent set and a minimum dominating set that are
disjoint and we show that the corresponding decision problem is NP-hard for
general graphs. Additionally, we prove several structural and hardness
results concerning pairs of disjoint sets in graphs which are dominating,
independent, or both. Furthermore, we prove lower bounds for the maximum
cardinality of an independent set of graphs with specifed odd girth and
small average degree.A connected graph G has spanning tree congestion at
most s, if G has a spanning tree T such that for every edge e of T the edge
cut defined in G by the vertex sets of the two components of T-e contains
at most s edges. We prove that every connected graph of order n has
spanning tree congestion at most n^(3/2) and we show that the corresponding
decision problem is NP-hard
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
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