2,383 research outputs found
Hitting all maximum cliques with a stable set using lopsided independent transversals
Rabern recently proved that any graph with omega >= (3/4)(Delta+1) contains a
stable set meeting all maximum cliques. We strengthen this result, proving that
such a stable set exists for any graph with omega > (2/3)(Delta+1). This is
tight, i.e. the inequality in the statement must be strict. The proof relies on
finding an independent transversal in a graph partitioned into vertex sets of
unequal size.Comment: 7 pages. v4: Correction to statement of Lemma 8 and clarified proof
A note on coloring vertex-transitive graphs
We prove bounds on the chromatic number of a vertex-transitive graph
in terms of its clique number and maximum degree . We
conjecture that every vertex-transitive graph satisfies and we
prove results supporting this conjecture. Finally, for vertex-transitive graphs
with we prove the Borodin-Kostochka conjecture, i.e.,
Exploiting -Closure in Kernelization Algorithms for Graph Problems
A graph is c-closed if every pair of vertices with at least c common
neighbors is adjacent. The c-closure of a graph G is the smallest number such
that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated
it in the context of clique enumeration. We show that c-closure can be applied
in kernelization algorithms for several classic graph problems. We show that
Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a
kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with
O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed
graphs have polynomially-bounded Ramsey numbers, as we show
A different short proof of Brooks' theorem
Lov\'asz gave a short proof of Brooks' theorem by coloring greedily in a good
order. We give a different short proof by reducing to the cubic case. Then we
show how to extend the result to (online) list coloring via the Kernel Lemma.Comment: added cute Kernel Lemma trick to lift up to (online) list colorin
The Graph Motif problem parameterized by the structure of the input graph
The Graph Motif problem was introduced in 2006 in the context of biological
networks. It consists of deciding whether or not a multiset of colors occurs in
a connected subgraph of a vertex-colored graph. Graph Motif has been mostly
analyzed from the standpoint of parameterized complexity. The main parameters
which came into consideration were the size of the multiset and the number of
colors. Though, in the many applications of Graph Motif, the input graph
originates from real-life and has structure. Motivated by this prosaic
observation, we systematically study its complexity relatively to graph
structural parameters. For a wide range of parameters, we give new or improved
FPT algorithms, or show that the problem remains intractable. For the FPT
cases, we also give some kernelization lower bounds as well as some ETH-based
lower bounds on the worst case running time. Interestingly, we establish that
Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which
is, to the best of our knowledge, the first problem to behave this way.Comment: 24 pages, accepted in DAM, conference version in IPEC 201
Bidimensionality and Geometric Graphs
In this paper we use several of the key ideas from Bidimensionality to give a
new generic approach to design EPTASs and subexponential time parameterized
algorithms for problems on classes of graphs which are not minor closed, but
instead exhibit a geometric structure. In particular we present EPTASs and
subexponential time parameterized algorithms for Feedback Vertex Set, Vertex
Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk
graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk
graphs. Our results are based on the recent decomposition theorems proved by
Fomin et al [SODA 2011], and our algorithms work directly on the input graph.
Thus it is not necessary to compute the geometric representations of the input
graph. To the best of our knowledge, these results are previously unknown, with
the exception of the EPTAS and a subexponential time parameterized algorithm on
unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and
Alber and Fiala [J. Algorithms 2004], respectively.
We proceed to show that our approach can not be extended in its full
generality to more general classes of geometric graphs, such as intersection
graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex
Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor
subexponential time algorithms unless the Exponential Time Hypothesis fails.
Additionally, we show that the decomposition theorems which our approach is
based on fail for disk graphs and that therefore any extension of our results
to disk graphs would require new algorithmic ideas. On the other hand, we prove
that our EPTASs and subexponential time algorithms for Vertex Cover and
Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs
in R^d for every fixed d
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