136 research outputs found
List homomorphism problems for signed graphs
We consider homomorphisms of signed graphs from a computational perspective.
In particular, we study the list homomorphism problem seeking a homomorphism of
an input signed graph , equipped with lists , of allowed images, to a fixed target signed graph . The
complexity of the similar homomorphism problem without lists (corresponding to
all lists being ) has been previously classified by Brewster and
Siggers, but the list version remains open and appears difficult. We illustrate
this difficulty by classifying the complexity of the problem when is a tree
(with possible loops). The tools we develop will be useful for classifications
of other classes of signed graphs, and we illustrate this by classifying the
complexity of irreflexive signed graphs in which the unicoloured edges form
some simple structures, namely paths or cycles. The structure of the signed
graphs in the polynomial cases is interesting, suggesting they may constitute a
nice class of signed graphs analogous to the so-called bi-arc graphs (which
characterize the polynomial cases of list homomorphisms to unsigned graphs).Comment: various changes + rewritten section on path- and cycle-separable
graphs based on a new conference submission (split possible in future
Complexity of planar signed graph homomorphisms to cycles
We study homomorphism problems of signed graphs. A signed graph is an
undirected graph where each edge is given a sign, positive or negative. An
important concept for signed graphs is the operation of switching at a vertex,
which is to change the sign of each incident edge. A homomorphism of a graph is
a vertex-mapping that preserves the adjacencies; in the case of signed graphs,
we also preserve the edge-signs. Special homomorphisms of signed graphs, called
s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the
mapping, to perform any number of switchings on the source signed graph. This
concept has been extensively studied, and a full complexity classification
(polynomial or NP-complete) for s-homomorphism to a fixed target signed graph
has recently been obtained. Such a dichotomy is not known when we restrict the
input graph to be planar (not even for non-signed graph homomorphisms).
We show that deciding whether a (non-signed) planar graph admits a
homomorphism to the square of a cycle with , or to the circular
clique with , are NP-complete problems. We use these
results to show that deciding whether a planar signed graph admits an
s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is
unbalanced if it has an odd number of negative edges). We deduce a complete
complexity dichotomy for the planar s-homomorphism problem with any signed
cycle as a target.
We also study further restrictions involving the maximum degree and the girth
of the input signed graph. We prove that planar s-homomorphism problems to
signed cycles remain NP-complete even for inputs of maximum degree~ (except
for the case of unbalanced -cycles, for which we show this for maximum
degree~). We also show that for a given integer , the problem for signed
bipartite planar inputs of girth is either trivial or NP-complete.Comment: 17 pages, 10 figure
List Homomorphism Problems for Signed Graphs
We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph (G,?), equipped with lists L(v) ? V(H), v ? V(G), of allowed images, to a fixed target signed graph (H,?). The complexity of the similar homomorphism problem without lists (corresponding to all lists being L(v) = V(H)) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. Both versions (with lists or without lists) can be formulated as constraint satisfaction problems, and hence enjoy the algebraic dichotomy classification recently verified by Bulatov and Zhuk. By contrast, we seek a combinatorial classification for the list version, akin to the combinatorial classification for the version without lists completed by Brewster and Siggers. We illustrate the possible complications by classifying the complexity of the list homomorphism problem when H is a (reflexive or irreflexive) signed tree. It turns out that the problems are polynomial-time solvable for certain caterpillar-like trees, and are NP-complete otherwise. The tools we develop will be useful for classifications of other classes of signed graphs, and we mention some follow-up research of this kind; those classifications are surprisingly complex
Parameterized complexity of edge-coloured and signed graph homomorphism problems
We study the complexity of graph modification problems for homomorphism-based
properties of edge-coloured graphs. A homomorphism from an edge-coloured graph
to an edge-coloured graph is a vertex-mapping from to that
preserves adjacencies and edge-colours. We consider the property of having a
homomorphism to a fixed edge-coloured graph . Given an edge-coloured graph
, can we perform graph operations so that the resulting graph has a
homomorphism to ? The operations we consider are vertex-deletion,
edge-deletion and switching (an operation that permutes the colours of the
edges incident to a given vertex). Switching plays an important role in the
theory of signed graphs, that are -edge-coloured graphs whose colours are
and . We denote the corresponding problems (parameterized by ) by
VERTEX DELETION -COLOURING, EDGE DELETION -COLOURING and SWITCHING
-COLOURING. These generalise -COLOURING (where one has to decide if an
input graph admits a homomorphism to ). Our main focus is when has order
at most , a case that includes standard problems such as VERTEX COVER, ODD
CYCLE TRANSVERSAL and EDGE BIPARTIZATION. For such a graph , we give a
P/NP-complete complexity dichotomy for all three studied problems. Then, we
address their parameterized complexity. We show that all VERTEX DELETION
-COLOURING and EDGE DELETION -COLOURING problems for such are FPT.
This is in contrast with the fact that already for some of order~,
unless P=NP, none of the three considered problems is in XP. We show that the
situation is different for SWITCHING -COLOURING: there are three
-edge-coloured graphs of order for which this is W-hard, and
assuming the ETH, admits no algorithm in time for inputs of size
. For the other cases, SWITCHING -COLOURING is FPT.Comment: 18 pages, 8 figures, 1 table. To appear in proceedings of IPEC 201
Graph modification for edge-coloured and signed graph homomorphism problems: parameterized and classical complexity
We study the complexity of graph modification problems with respect to
homomorphism-based colouring properties of edge-coloured graphs. A homomorphism
from edge-coloured graph to edge-coloured graph is a vertex-mapping
from to that preserves adjacencies and edge-colours. We consider the
property of having a homomorphism to a fixed edge-coloured graph . The
question we are interested in is: given an edge-coloured graph , can we
perform graph operations so that the resulting graph admits a homomorphism
to ? The operations we consider are vertex-deletion, edge-deletion and
switching (an operation that permutes the colours of the edges incident to a
given vertex). Switching plays an important role in the theory of signed
graphs, that are 2-edge-coloured graphs whose colours are the signs and
. We denote the corresponding problems (parameterized by ) by
VD--COLOURING, ED--COLOURING and SW--COLOURING. These problems
generalise -COLOURING (to decide if an input graph admits a homomorphism to
a fixed target ).
Our main focus is when is an edge-coloured graph with at most two
vertices, a case that is already interesting as it includes problems such as
VERTEX COVER, ODD CYCLE RANSVERSAL and EDGE BIPARTIZATION. For such a graph
, we give a P/NP-c complexity dichotomy for VD--COLOURING,
ED--COLOURING and SW--COLOURING. We then address their parameterized
complexity. We show that VD--COLOURING and ED--COLOURING for all such
are FPT. In contrast, already for some of order 3, unless P=NP, none of the
three problems is in XP, since 3-COLOURING is NP-c. We show that
SW--COLOURING is different: there are three 2-edge-coloured graphs of
order 2 for which SW--COLOURING is W-hard, and assuming the ETH, admits no
algorithm in time . For the other cases, SW--COLOURING is FPT.Comment: 17 pages, 9 figures, 2 table
A dichotomy theorem for -switchable -colouring on -edge coloured graphs
Let be a graph in which each edge is assigned one of the colours , and let be a subgroup of . The operation of switching
at a vertex of with respect to an element of permutes
the colours of the edges incident with according to . We investigate
the complexity of whether there exists a sequence of switches that transforms a
given -edge coloured graph so that it has a colour-preserving
homomorphism to a fixed -edge coloured graph and give a dichotomy
theorem in the case that acts transitively.Comment: 14 pages, 2 figure
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