1,617 research outputs found
The complexity of partitioning into disjoint cliques and a triangle-free graph
Motivated by Chudnovsky's structure theorem of bull-free graphs, Abu-Khzam,
Feghali, and M\"uller have recently proved that deciding if a graph has a
vertex partition into disjoint cliques and a triangle-free graph is NP-complete
for five graph classes. The problem is trivial for the intersection of these
five classes. We prove that the problem is NP-complete for the intersection of
two subsets of size four among the five classes. We also show NP-completeness
for other small classes, such as graphs with maximum degree 4 and line graphs
A semi-algebraic version of Zarankiewicz's problem
A bipartite graph is semi-algebraic in if its vertices are
represented by point sets and its edges are defined
as pairs of points that satisfy a Boolean combination of
a fixed number of polynomial equations and inequalities in coordinates. We
show that for fixed , the maximum number of edges in a -free
semi-algebraic bipartite graph in with
and is at most , and this bound is tight. In
dimensions , we show that all such semi-algebraic graphs have at most
edges, where here
is an arbitrarily small constant and .
This result is a far-reaching generalization of the classical
Szemer\'edi-Trotter incidence theorem. The proof combines tools from several
fields: VC-dimension and shatter functions, polynomial partitioning, and
Hilbert polynomials.
We also present various applications of our theorem. For example, a general
point-variety incidence bound in , an improved bound for a
-dimensional variant of the Erd\H{o}s unit distances problem, and more
Complexity of choosability with a small palette of colors
A graph is -choosable if, for any choice of lists of colors for
each vertex, there is a list coloring, which is a coloring where each vertex
receives a color from its list. We study complexity issues of choosability of
graphs when the number of colors is limited. We get results which differ
surprisingly from the usual case where is implicit and which extend known
results for the usual case. We also exhibit some classes of graphs (defined by
structural properties of their blocks) which are choosable.Comment: 31 pages, 11 figure
Parameterized complexity of the spanning tree congestion problem
We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k≤3 the problem becomes polynomially time solvable.publishedVersio
Graph Isomorphism Restricted by Lists
The complexity of graph isomorphism (GraphIso) is a famous unresolved problem
in theoretical computer science. For graphs and , it asks whether they
are the same up to a relabeling of vertices. In 1981, Lubiw proved that list
restricted graph isomorphism (ListIso) is NP-complete: for each ,
we are given a list of possible images of
. After 35 years, we revive the study of this problem and consider which
results for GraphIso translate to ListIso.
We prove the following: 1) When GraphIso is GI-complete for a class of
graphs, it translates into NP-completeness of ListIso. 2) Combinatorial
algorithms for GraphIso translate into algorithms for ListIso: for trees,
planar graphs, interval graphs, circle graphs, permutation graphs, bounded
genus graphs, and bounded treewidth graphs. 3) Algorithms based on group theory
do not translate: ListIso remains NP-complete for cubic colored graphs with
sizes of color classes bounded by 8.
Also, ListIso allows to classify results for the graph isomorphism problem.
Some algorithms are robust and translate to ListIso. A fundamental problem is
to construct a combinatorial polynomial-time algorithm for cubic graph
isomorphism, avoiding group theory. By the 3rd result, ListIso is NP-hard for
them, so no robust algorithm for cubic graph isomorphism exists, unless P = NP
Chromatic roots are dense in the whole complex plane
I show that the zeros of the chromatic polynomials P-G(q) for the generalized theta graphs Theta((s.p)) are taken together, dense in the whole complex plane with the possible exception of the disc \q - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof
On the approximability of the maximum induced matching problem
In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3
The VC-Dimension of Graphs with Respect to k-Connected Subgraphs
We study the VC-dimension of the set system on the vertex set of some graph
which is induced by the family of its -connected subgraphs. In particular,
we give tight upper and lower bounds for the VC-dimension. Moreover, we show
that computing the VC-dimension is -complete and that it remains
-complete for split graphs and for some subclasses of planar
bipartite graphs in the cases and . On the positive side, we
observe it can be decided in linear time for graphs of bounded clique-width
On the Connectivity Preserving Minimum Cut Problem
In this paper, we study a generalization of the classical minimum cut prob-
lem, called Connectivity Preserving Minimum Cut (CPMC) problem, which seeks a
minimum cut to separate a pair (or pairs) of source and destination nodes and
meanwhile ensure the connectivity between the source and its partner node(s).
The CPMC problem is a rather powerful formulation for a set of problems and
finds applications in many other areas, such as network security, image
processing, data mining, pattern recognition, and machine learning. For this
important problem, we consider two variants, connectiv- ity preserving minimum
node cut (CPMNC) and connectivity preserving minimum edge cut (CPMEC). For
CPMNC, we show that it cannot be ap- proximated within {\alpha}logn for some
constant {\alpha} unless P=NP, and cannot be approximated within any poly(logn)
unless NP has quasi-polynomial time algorithms. The hardness results hold even
for graphs with unit weight and bipartite graphs. Particularly, we show that
polynomial time solutions exist for CPMEC in planar graphs and for CPMNC in
some special planar graphs. The hardness of CPMEC in general graphs remains
open, but the polynomial time algorithm in planar graphs still has important
practical applications
Parameterized Complexity of Diameter
Diameter -- the task of computing the length of a longest shortest path -- is
a fundamental graph problem. Assuming the Strong Exponential Time Hypothesis,
there is no -time algorithm even in sparse graphs [Roditty and
Williams, 2013]. To circumvent this lower bound we aim for algorithms with
running time where is a parameter and is a function as
small as possible. We investigate which parameters allow for such running
times. To this end, we systematically explore a hierarchy of structural graph
parameters
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