Symmetric-Difference (Degeneracy) and Signed Tree Models

We introduce a dense counterpart of graph degeneracy, which extends the recently-proposed invariant symmetric difference. We say that a graph has sd-degeneracy (for symmetric-difference degeneracy) at most d if it admits an elimination order of its vertices where a vertex u can be removed whenever it has a d-twin, i.e., another vertex v such that at most d vertices outside {u,v} are neighbors of exactly one of u, v. The family of graph classes of bounded sd-degeneracy is a superset of that of graph classes of bounded degeneracy or of bounded flip-width, and more generally, of bounded symmetric difference. Unlike most graph parameters, sd-degeneracy is not hereditary: it may be strictly smaller on a graph than on some of its induced subgraphs. In particular, every n-vertex graph is an induced subgraph of some O(n²)-vertex graph of sd-degeneracy 1. In spite of this and the breadth of classes of bounded sd-degeneracy, we devise Õ(√n)-bit adjacency labeling schemes for them, which are optimal up to the hidden polylogarithmic factor. This is attained on some even more general classes, consisting of graphs G whose vertices bijectively map to the leaves of a tree T, where transversal edges and anti-edges added to T define the edge set of G. We call such graph representations signed tree models as they extend the so-called tree models (or twin-decompositions) developed in the context of twin-width, by adding transversal anti-edges. While computing the degeneracy of a graph takes linear time, we show that determining its symmetric difference is para-co-NP-complete. This may seem surprising as symmetric difference can serve as a short-sighted first approximation of twin-width, whose computation is para-NP-complete. Indeed, we show that deciding if the symmetric difference of an input graph is at most 8 is co-NP-complete. We also show that deciding if the sd-degeneracy is at most 6 is NP-complete, contrasting with the symmetric difference.