Constant-Size Certificates for Leader Election in Chordal Graphs and Related Classes

TL;DR

The paper tackles verifying leader election and spanning-tree solutions in distributed graphs using constant-size certificates per edge within a local certification framework. It exploits graph orderings—simplicial elimination for chordal graphs and dismantling orders for dismantlable graphs—to enable local checks that guarantee a unique sink (leader) and, in chordal graphs, an acyclic orientation. It delivers constant-size certification schemes for leader election in chordal and $K_4$-free dismantlable graphs, plus a spanning-tree certification for dismantlable graphs with a designated root, with the chordal scheme additionally enforcing acyclicity. Furthermore, it shows how any certificate can be transformed into a silent self-stabilizing algorithm under a Gouda fair scheduler by adding only a single extra state, advancing fault-tolerance with minimal memory in natural graph classes.

Abstract

In distributed computing a certification scheme consists of a set of states and conditions over those states that enable each node of a graph to efficiently verify the correctness of a solution to a given problem. This work focuses on two fundamental problems: leader election and spanning tree construction. For each problem, we present a constant-size (per edge), local certification scheme, where the conditions available to each node can only refer to the graph induced by its one-hop neighborhood. In particular, we provide certification schemes for leader election in chordal and $K_4$-free dismantlable graphs and for spanning tree construction in dismantlable graphs, assuming a root is given. For chordal graphs, our leader election certification scheme additionally ensures an acyclic orientation, a property that is not generally verifiable using constant-size certificates in arbitrary graphs. To the best of our knowledge, these are the first local certification results tailored to these graph classes, potentially highlighting structural properties useful for verifying additional problems. Finally, we propose an algorithm that automatically transforms any certification scheme into a silent self-stabilizing algorithm (i.e., an algorithm that automatically recovers from faults) by adding only one extra state to the set of states of the certification scheme, assuming a Gouda fair scheduler. This transformation may be of independent interest.