2,079 research outputs found
Transitivity conditions in infinite graphs
We study transitivity properties of graphs with more than one end. We
completely classify the distance-transitive such graphs and, for all , the -CS-transitive such graphs.Comment: 20 page
Clique trees of infinite locally finite chordal graphs
We investigate clique trees of infinite locally finite chordal graphs. Our
main contribution is a bijection between the set of clique trees and the
product of local finite families of finite trees. Even more, the edges of a
clique tree are in bijection with the edges of the corresponding collection of
finite trees. This allows us to enumerate the clique trees of a chordal graph
and extend various classic characterisations of clique trees to the infinite
setting
Polynomial-time algorithm for Maximum Weight Independent Set on -free graphs
In the classic Maximum Weight Independent Set problem we are given a graph
with a nonnegative weight function on vertices, and the goal is to find an
independent set in of maximum possible weight. While the problem is NP-hard
in general, we give a polynomial-time algorithm working on any -free
graph, that is, a graph that has no path on vertices as an induced
subgraph. This improves the polynomial-time algorithm on -free graphs of
Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on
-free graphs of Lokshtanov et al (SODA 2016). The main technical
contribution leading to our main result is enumeration of a polynomial-size
family of vertex subsets with the following property: for every
maximal independent set in the graph, contains all maximal
cliques of some minimal chordal completion of that does not add any edge
incident to a vertex of
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
Homogeneous sets, clique-separators, critical graphs, and optimal -binding functions
Given a set of graphs, let be the optimal -binding function of
the class of -free graphs, that is,
In this paper, we combine the
two decomposition methods by homogeneous sets and clique-separators in order to
determine optimal -binding functions for subclasses of -free graphs
and of -free graphs. In particular, we prove the following
for each :
(i)
(ii) $\
f_{\{P_5,co-banner\}}^\star(\omega)=f^\star_{\{2K_2\}}(\omega)\in\mathcal{O}(\omega^2),\
f_{\{C_5,C_7,\ldots,banner\}}^\star(\omega)=f^\star_{\{C_5,3K_1\}}(\omega)\notin
\mathcal{O}(\omega),\ f_{\{P_5,C_4\}}^\star(\omega)=\lceil(5\omega-1)/4\rceil.G\chi(G)>\chi(G-u)u\in V(G)(P_5,banner)$-free graphs
Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
Given a directed graph , a set of terminals and an integer , the
\textsc{Directed Vertex Multiway Cut} problem asks if there is a set of at
most (nonterminal) vertices whose removal disconnects each terminal from
all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous
problem where is a set of at most edges. These two problems indeed are
known to be equivalent. A natural generalization of the multiway cut is the
\emph{multicut} problem, in which we want to disconnect only a set of given
pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in
undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized
by . Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and
directed multicut is W[1]-hard parameterized by . We complete the picture
here by our main result which is that both \textsc{Directed Vertex Multiway
Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time
, i.e., FPT parameterized by size of the cutset of
the solution. This answers an open question raised by Marx (Theor. Comp. Sci.
2006) and Marx and Razgon (STOC 2011). It follows from our result that
\textsc{Directed Multicut} is FPT for the case of terminal pairs, which
answers another open problem raised in Marx and Razgon (STOC 2011)
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