A dominating set of a graph G = (V,E) is a set of vertices D ⊆ V whose closed neighborhood is V, i.e., N[D] = V. We view a dominating set as a collection of tokens placed on the vertices of D. In the token sliding variant of the Dominating Set Reconfiguration problem (TS-DSR), we seek to transform a source dominating set into a target dominating set in G by sliding tokens along edges, and while maintaining a dominating set all along the transformation. TS-DSR is known to be PSPACE-complete even restricted to graphs of pathwidth w, for some non-explicit constant w and to be XL-complete parameterized by the size k of the solution. The first contribution of this article consists in using a novel approach to provide the first explicit constant for which the TS-DSR problem is PSPACE-complete, a question that was left open in the literature. From a parameterized complexity perspective, the token jumping variant of DSR, i.e., where tokens can jump to arbitrary vertices, is known to be FPT when parameterized by the size of the dominating sets on nowhere dense classes of graphs. But, in contrast, no non-trivial result was known about TS-DSR. We prove that DSR is actually much harder in the sliding model since it is XL-complete when restricted to bounded pathwidth graphs and even when parameterized by k plus the feedback vertex set number of the graph. This gives, for the first time, a difference of behavior between the complexity under token sliding and token jumping for some problem on graphs of bounded treewidth. All our results are obtained using a brand new method, based on the hardness of the so-called Tape Reconfiguration problem, a problem we believe to be of independent interest. We complement these hardness results with a positive result showing that DSR (parameterized by k) in the sliding model is FPT on planar graphs, also answering an open problem from the literature.

Abstract. The discharging method is a powerful proof technique, especially for graph coloring problems. Its major downside is that it often requires lengthy case analyses, which are sometimes given to a computer for verification. However, it is much less common to use a computer to actively look for a discharging proof. In this paper, we use a linear programming approach to automatically look for a discharging proof. While our system is not entirely autonomous, we manage to make some progress toward Wegner’s conjecture for distance-2 coloring of planar graphs by showing that 12 colors are sufficient to color at distance 2 every planar graph with maximum degree 4.

Abstract. For an integer [Formula: see text] and a graph [Formula: see text], let [Formula: see text] be the graph that has vertex set all proper [Formula: see text]-colorings of [Formula: see text], and an edge between two vertices [Formula: see text] and [Formula: see text] whenever the coloring [Formula: see text] can be obtained from [Formula: see text] by a single Kempe change. A theorem of Meyniel from 1978 states that [Formula: see text] is connected with diameter [Formula: see text] for every planar graph [Formula: see text]. We significantly strengthen this result by showing that there is a positive constant [Formula: see text] such that [Formula: see text] has diameter [Formula: see text] for every planar graph [Formula: see text].