16,207 research outputs found
Vertex Cover Gets Faster and Harder on Low Degree Graphs
The problem of finding an optimal vertex cover in a graph is a classic
NP-complete problem, and is a special case of the hitting set question. On the
other hand, the hitting set problem, when asked in the context of induced
geometric objects, often turns out to be exactly the vertex cover problem on
restricted classes of graphs. In this work we explore a particular instance of
such a phenomenon. We consider the problem of hitting all axis-parallel slabs
induced by a point set P, and show that it is equivalent to the problem of
finding a vertex cover on a graph whose edge set is the union of two
Hamiltonian Paths. We show the latter problem to be NP-complete, and we also
give an algorithm to find a vertex cover of size at most k, on graphs of
maximum degree four, whose running time is 1.2637^k n^O(1)
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
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