Symmetric-Difference (Degeneracy) and Signed Tree Models

TL;DR

The paper develops sd-degeneracy as a dense counterpart to graph degeneracy, defined via the existence of a vertex ordering where each vertex has a d-twin in the remaining suffix. It introduces signed tree models to capture sd-degeneracy structurally, showing that graphs with sd-degeneracy d admit clean signed tree models of width at most d+1, and provides a depth-balancing procedure to produce efficient adjacency labeling schemes with near-optimal size, specifically $ ilde{O}(\,\sqrt{d n})$ up to polylog factors. The authors prove strong complexity results: deciding sd-degeneracy ≤ 1 is NP-complete, while deciding symmetric difference ≤ 8 is co-NP-complete (and para-co-NP for fixed d), highlighting a nuanced difference between these two dense parameters. They also connect these structural insights to practical implicit representations and labeling schemes for broad graph classes, while clarifying the non-hereditary nature of sd-degeneracy and its inclusivity over bounded degeneracy and bounded symmetric difference classes. The work thus advances the theoretical understanding of dense graph parameters and their algorithmic implications, including explicit gadget-based reductions linking logical satisfiability to graph structural properties.

Abstract

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 $O(n^2)$-vertex graph of sd-degeneracy 1. In spite of this and the breadth of classes of bounded sd-degeneracy, we devise $\tilde{O}(\sqrt n)$-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 an input graph can be done in linear time, we show that deciding whether its symmetric difference is at most 8 is co-NP-complete, and whether its sd-degeneracy is at most 1 is NP-complete.

Figures (7)

• Figure 1: A signed tree model of a 14-vertex graph.
• Figure 2: The signed tree model of \ref{['fig:signed-tree-model']} made clean.
• Figure 3: A neatly $(2,2,2,7)$-attached bubble gadget, with $d=12$. The vertical and horizontal red boxes are cliques.
• Figure 4: The variable gadget of $x$ with $d=12$.
• Figure 5: The essential part of $G$ built so far, for a 3-CNF formula $\varphi$ whose first two clauses are $x_1 \lor \neg x_3 \lor x_4$ and $\neg x_2 \lor x_3 \lor \neg x_4$. The blue ellipses represent the bubbles attached to the four enclosed vertices (recall that the bubble is attached to a fifth vertex among the sets $N_x$).

Theorems & Definitions (36)

• Proposition 1.1
• proof
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 3.1
• proof
• Claim 3.2
• proof : Proof of Claim.
• Proposition 4.1