Leader Election via Unique Sink Orientation

A Locally Checkable Labeling (LCL) is a specification describing a set of labels that are valid with respect to a set of conditions that characterize a local part of a solution to a global problem. Conditions can only refer to nodes and labels within a constant radius neighborhood of each node. This work studies local labeling schemes whose global consistency implies solutions to two classical problems: leader election and spanning tree construction. For each problem, we present a local labeling scheme using one bit per edge or equivalently $O(Δ)$ bits per node (where $Δ$ is the maximum degree in the graph), with conditions checkable within the graph induced by the one neighborhood of each node. For leader election, we show that global satisfaction of the conditions implies the existence of a unique sink in the graph, which we define to be a leader, while in the spanning tree setting it implies that a specific subset of edges induces a spanning tree rooted at a given node. We show those implications for $K_4$-free dismantlable and chordal graphs in the former case and for dismantlable graphs in the latter, assuming a root is given. For chordal graphs, the labeling implying a unique sink additionally induces an acyclic orientation. This property is not generally locally verifiable with constant-size labels in arbitrary graphs. To the best of our knowledge, these are the first local labeling results tailored to ($K_4$-free) dismantlable graphs, potentially highlighting structural properties useful for designing LCLs for additional problems. Finally, we present a generic transformation that converts any local labeling scheme into a silent self-stabilizing algorithm by adding only one extra state, assuming a Gouda fair scheduler. This transformation may be of independent interest.