30,443 research outputs found
Computing and counting longest paths on circular-arc graphs in polynomial time.
The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O(n4)) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately âcutâ the circle, such that the obtained (not induced) interval subgraph GⲠof G admits a path PⲠon the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O(n4) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight
Computing and Counting Longest Paths on Circular-Arc Graphs in Polynomial Time
The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O(n4)) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately âcutâ the circle, such that the obtained (not induced) interval subgraph GⲠof G admits a path PⲠon the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O(n4) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight
Efficient algorithms for tuple domination on co-biconvex graphs and web graphs
A vertex in a graph dominates itself and each of its adjacent vertices. The
-tuple domination problem, for a fixed positive integer , is to find a
minimum sized vertex subset in a given graph such that every vertex is
dominated by at least k vertices of this set. From the computational point of
view, this problem is NP-hard. For a general circular-arc graph and ,
efficient algorithms are known to solve it (Hsu et al., 1991 & Chang, 1998) but
its complexity remains open for . A -matrix has the consecutive
0's (circular 1's) property for columns if there is a permutation of its rows
that places the 0's (1's) consecutively (circularly) in every column.
Co-biconvex (concave-round) graphs are exactly those graphs whose augmented
adjacency matrix has the consecutive 0's (circular 1's) property for columns.
Due to A. Tucker (1971), concave-round graphs are circular-arc. In this work,
we develop a study of the -tuple domination problem on co-biconvex graphs
and on web graphs which are not comparable and, in particular, all of them
concave-round graphs. On the one side, we present an -time algorithm
for solving it for each , where is the set of universal
vertices and the total number of vertices of the input co-biconvex graph.
On the other side, the study of this problem on web graphs was already started
by Argiroffo et al. (2010) and solved from a polyhedral point of view only for
the cases and , where equals the degree of each vertex of
the input web graph . We complete this study for web graphs from an
algorithmic point of view, by designing a linear time algorithm based on the
modular arithmetic for integer numbers. The algorithms presented in this work
are independent but both exploit the circular properties of the augmented
adjacency matrices of each studied graph class.Comment: 21 pages, 7 figures. Keywords: -tuple dominating sets, augmented
adjacency matrices, stable sets, modular arithmeti
Deciding Circular-Arc Graph Isomorphism in Parameterized Logspace
We compute a canonical circular-arc representation for a given circular-arc
(CA) graph which implies solving the isomorphism and recognition problem for
this class. To accomplish this we split the class of CA graphs into uniform and
non-uniform ones and employ a generalized version of the argument given by
K\"obler et al (2013) that has been used to show that the subclass of Helly CA
graphs can be canonized in logspace. For uniform CA graphs our approach works
in logspace and in addition to that Helly CA graphs are a strict subset of
uniform CA graphs. Thus our result is a generalization of the canonization
result for Helly CA graphs. In the non-uniform case a specific set of ambiguous
vertices arises. By choosing the parameter to be the cardinality of this set
the obstacle can be solved by brute force. This leads to an O(k + log n) space
algorithm to compute a canonical representation for non-uniform and therefore
all CA graphs.Comment: 14 pages, 3 figure
Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space
We present a logspace algorithm that constructs a canonical intersection
model for a given proper circular-arc graph, where `canonical' means that
models of isomorphic graphs are equal. This implies that the recognition and
the isomorphism problems for this class of graphs are solvable in logspace. For
a broader class of concave-round graphs, that still possess (not necessarily
proper) circular-arc models, we show that those can also be constructed
canonically in logspace. As a building block for these results, we show how to
compute canonical models of circular-arc hypergraphs in logspace, which are
also known as matrices with the circular-ones property. Finally, we consider
the search version of the Star System Problem that consists in reconstructing a
graph from its closed neighborhood hypergraph. We solve it in logspace for the
classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio
Efficient and Perfect domination on circular-arc graphs
Given a graph , a \emph{perfect dominating set} is a subset of
vertices such that each vertex is
dominated by exactly one vertex . An \emph{efficient dominating set}
is a perfect dominating set where is also an independent set. These
problems are usually posed in terms of edges instead of vertices. Both
problems, either for the vertex or edge variant, remains NP-Hard, even when
restricted to certain graphs families. We study both variants of the problems
for the circular-arc graphs, and show efficient algorithms for all of them
Uniquely D-colourable digraphs with large girth
Let C and D be digraphs. A mapping is a C-colouring if for
every arc of D, either is an arc of C or , and the
preimage of every vertex of C induces an acyclic subdigraph in D. We say that D
is C-colourable if it admits a C-colouring and that D is uniquely C-colourable
if it is surjectively C-colourable and any two C-colourings of D differ by an
automorphism of C. We prove that if a digraph D is not C-colourable, then there
exist digraphs of arbitrarily large girth that are D-colourable but not
C-colourable. Moreover, for every digraph D that is uniquely D-colourable,
there exists a uniquely D-colourable digraph of arbitrarily large girth. In
particular, this implies that for every rational number , there are
uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of
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