21,612 research outputs found
On Minimum Maximal Distance-k Matchings
We study the computational complexity of several problems connected with
finding a maximal distance- matching of minimum cardinality or minimum
weight in a given graph. We introduce the class of -equimatchable graphs
which is an edge analogue of -equipackable graphs. We prove that the
recognition of -equimatchable graphs is co-NP-complete for any fixed . We provide a simple characterization for the class of strongly chordal
graphs with equal -packing and -domination numbers. We also prove that
for any fixed integer the problem of finding a minimum weight
maximal distance- matching and the problem of finding a minimum weight
-independent dominating set cannot be approximated in polynomial
time in chordal graphs within a factor of unless
, where is a fixed constant (thereby
improving the NP-hardness result of Chang for the independent domination case).
Finally, we show the NP-hardness of the minimum maximal induced matching and
independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure
On (t,r) Broadcast Domination Numbers of Grids
The domination number of a graph is the minimum cardinality of
any subset such that every vertex in is in or adjacent to
an element of . Finding the domination numbers of by grids was an
open problem for nearly 30 years and was finally solved in 2011 by Goncalves,
Pinlou, Rao, and Thomass\'e. Many variants of domination number on graphs have
been defined and studied, but exact values have not yet been obtained for
grids. We will define a family of domination theories parameterized by pairs of
positive integers where which generalize domination
and distance domination theories for graphs. We call these domination numbers
the broadcast domination numbers. We give the exact values of
broadcast domination numbers for small grids, and we identify upper bounds for
the broadcast domination numbers for large grids and conjecture that
these bounds are tight for sufficiently large grids.Comment: 28 pages, 43 figure
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