4 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 (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
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