Twin-width one

TL;DR

We determine the structure of graphs with twin-width at most $1$, proving they are permutation graphs and admit permutation-diagram realisers. We develop a linear-time recognition algorithm that combines modular decomposition with realiser-guided contraction sequencing to either produce a $1$-contraction sequence or certify width greater than $1$, using a recursive decomposition to guide the construction. We establish a close connection to distance-hereditary graphs, showing linear-time computation of optimal sequences for this class and providing a tight bound: twin-width is $0$ for cographs, $1$ for AT-free (i.e., permutation) distance-hereditary graphs, and $2$ otherwise. These results link twin-width $1$ to classical graph classes and support efficient FO-model-checking and related tasks on bounded-twin-width graphs.

Abstract

We investigate the structure of graphs of twin-width at most $1$, and obtain the following results: - Graphs of twin-width at most $1$ are permutation graphs. In particular they have an intersection model and a linear structure. - There is always a $1$-contraction sequence closely following a given permutation diagram. - Based on a recursive decomposition theorem, we obtain a simple algorithm running in linear time that produces a $1$-contraction sequence of a graph, or guarantees that it has twin-width more than $1$. - We characterise distance-hereditary graphs based on their twin-width and deduce a linear time algorithm to compute optimal sequences on this class of graphs.

Figures (4)

• Figure 1: The different cases of the induction constructing the realiser.
• Figure 2: An example of a twin-width 1 graph and its permutation diagram. Shown here is the decomposition of Lemma \ref{['lem:dec-algo']} starting from extremal vertex $s_1$. The segments of splitters $s_1,s_2,s_3$ are represented in dashed blue, green and orange respectively. Then, intervals of the corresponding colour are the intervals $A,B,C$ produced during the iteration in which the splitter is considered. $A$ and $B$ induce subgraphs $G^i$ from Lemma \ref{['lem:dec-tww1']}. Then, segments of the same colour as a splitter correspond to doubly extremal vertices eliminated in the corresponding step.
• Figure 3: Minimal graphs containing an asteroid triple.
• Figure 4: Complements of minimal graphs containing a $(2k+1)$-asteroid for $k>1$.

Theorems & Definitions (28)

• Lemma 2: Ahn, Hendrey, Kim, and Oum AhnHKO22
• Lemma 5: Bonnet, Kim, Reinald, Thomassé, and Watrigant BonnetKRTW22
• Corollary 6
• Lemma 7: Bonnet, Kim, Reinald, Thomassé, and Watrigant BonnetKRTW22
• Theorem 8
• Theorem 9
• Lemma 10
• Lemma 11
• Corollary 12
• Corollary 13