List Homomorphisms to Separable Signed Graphs

Abstract

The complexity of the list homomorphism problem for signed graphs appears difficult to classify. Existing results focus on special classes of signed graphs, such as trees [4] and reflexive signed graphs [25]. Irreflexive signed graphs are in a certain sense the heart of the problem, as noted by a recent paper of Kim and Siggers. We focus on a special class of irreflexive signed graphs, namely those in which the unicoloured edges form a spanning path or cycle, which we call separable signed graphs. We classify the complexity of list homomorphisms to these separable signed graphs; we believe that these signed graphs will play an important role for the general resolution of the irreflexive case. We also relate our results to a conjecture of Kim and Siggers concerning the special case of weakly balanced irreflexive signed graphs; we have proved the conjecture in another paper, and the present results add structural information to that topic.

Mapping sparse signed graphs to $(K_{2k}, M)$

“…Furthermore, we label the other neighbors of u ′ and v ′ as u 1 and v 1 , respectively, but noting that, as we will consider them to be precolored vertices, they do not need to be distinct from each other or from w ′ which will also be precolored. Then the set of admissible lists on u, v, w, induced by the coloring of u 1 , v 1 and w ′ , would satisfy the conditions of Lemma 4.11 (1) with k = 1, proving that this configuration is reducible.…”

“…v 2k+1 has one 3 1 -neighbor, v 2k+2 , which itself has a 3 1 -neighbor in G (P can be seen as one end of a poor path). Then in case i. if k ≥ 1, v 1 gives a charge of 1 2 to v 2 , in case ii. v 1 will give a charge of 1 2 to v 2 (that is even if k = 0).…”

“…Such extension problems lead to the idea of (K 6 , M)-list coloring or DSG(K 6 , M)-list coloring. We refer to [1] and references there for a general study of list-homomorphism problem of graphs and signed graphs. List assignments considered in this work, however, are rather restricted.…”

“…The problem then is reduced to a list coloring problem on (H, σ), but as (H, σ) has a unique cycle, we may further reduce the problem to list coloring of the cycles. However, in all but one of the cases we may apply Lemma 4.11 (1). In the exceptional case when v 0 is adjacent to v 2k+2 and v 1 is adjacent to v 2k+1 , we consider the 4-cycle v 0 v 2k+2 v 2k+1 v 1 and let (H, σ) be the subgraph induced by this 4-cycle and the two 2-neighbors of v 0 and v 2k+2 .…”

“…As before we consider the signed subgraph (H, σ) induced by the cycle and its 2-neighbors. Then again a coloring of (G, σ) − (H, σ) can be extended to (H, σ) by Lemma 4.11 (1).…”