229 research outputs found
On some third parts of nearly complete digraphs
AbstractFor the complete digraph DKn with n⩾3, its half as well as its third (or near-third) part, both non-self-converse, are exhibited. A backtracking method for generating a tth part of a digraph is sketched. It is proved that some self-converse digraphs are not among the near-third parts of the complete digraph DK5 in four of the six possible cases. For n=9+6k,k=0,1,…, a third part D of DKn is found such that D is a self-converse oriented graph and all D-decompositions of DKn have trivial automorphism group
A Linear Kernel for Planar Total Dominating Set
A total dominating set of a graph is a subset such
that every vertex in is adjacent to some vertex in . Finding a total
dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on
general graphs when parameterized by the solution size. By the meta-theorem of
Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total
Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how
such a kernel can be effectively constructed, and how to obtain explicit
reduction rules with reasonably small constants. Following the approach of
Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating
Set on planar graphs with at most vertices, where is the size of the
solution. This result complements several known constructive linear kernels on
planar graphs for other domination problems such as Dominating Set, Edge
Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue
Dominating Set.Comment: 33 pages, 13 figure
Clique-Stable Set separation in perfect graphs with no balanced skew-partitions
Inspired by a question of Yannakakis on the Vertex Packing polytope of
perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary
subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates
a clique K and a stable set S if and . A
Clique-Stable Set Separator is a family of cuts such that for every clique K,
and for every stable set S disjoint from K, there exists a cut in the family
that separates K and S. Given a class of graphs, the question is to know
whether every graph of the class admits a Clique-Stable Set Separator
containing only polynomially many cuts. It is open for the class of all graphs,
and also for perfect graphs, which was Yannakakis' original question. Here we
investigate on perfect graphs with no balanced skew-partition; the balanced
skew-partition was introduced in the proof of the Strong Perfect Graph Theorem.
Recently, Chudnovsky, Trotignon, Trunck and Vuskovic proved that forbidding
this unfriendly decomposition permits to recursively decompose Berge graphs
using 2-join and complement 2-join until reaching a basic graph, and they found
an efficient combinatorial algorithm to color those graphs. We apply their
decomposition result to prove that perfect graphs with no balanced
skew-partition admit a quadratic-size Clique-Stable Set Separator, by taking
advantage of the good behavior of 2-join with respect to this property. We then
generalize this result and prove that the Strong Erdos-Hajnal property holds in
this class, which means that every such graph has a linear-size biclique or
complement biclique. This property does not hold for all perfect graphs (Fox
2006), and moreover when the Strong Erdos-Hajnal property holds in a hereditary
class of graphs, then both the Erdos-Hajnal property and the polynomial
Clique-Stable Set Separation hold.Comment: arXiv admin note: text overlap with arXiv:1308.644
Sparsest Cut on Bounded Treewidth Graphs: Algorithms and Hardness Results
We give a 2-approximation algorithm for Non-Uniform Sparsest Cut that runs in
time , where is the treewidth of the graph. This improves on the
previous -approximation in time \poly(n) 2^{O(k)} due to
Chlamt\'a\v{c} et al.
To complement this algorithm, we show the following hardness results: If the
Non-Uniform Sparsest Cut problem has a -approximation for series-parallel
graphs (where ), then the Max Cut problem has an algorithm with
approximation factor arbitrarily close to . Hence, even for such
restricted graphs (which have treewidth 2), the Sparsest Cut problem is NP-hard
to approximate better than for ; assuming the
Unique Games Conjecture the hardness becomes . For
graphs with large (but constant) treewidth, we show a hardness result of assuming the Unique Games Conjecture.
Our algorithm rounds a linear program based on (a subset of) the
Sherali-Adams lift of the standard Sparsest Cut LP. We show that even for
treewidth-2 graphs, the LP has an integrality gap close to 2 even after
polynomially many rounds of Sherali-Adams. Hence our approach cannot be
improved even on such restricted graphs without using a stronger relaxation
Combinatorial properties of the G-degree
A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the "G-degree" of the involved graphs, which drives the 1/N expansion in the tensor models context. In the present paper - by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph - we prove that, in any even dimension d greater or equal to 4, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of (d-1)!. As a consequence, in even dimension, the terms of the 1/N expansion corresponding to odd powers of 1/N are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context. In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular genera with respect to an arbitrary pair of "associated" cyclic permutations, several results are obtained, relating the G-degree or the regular genus of 5-colored graphs and the Euler characteristic of the associated PL 4-manifolds
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