Bounds for the Twin-width of Graphs

Abstract

Bonnet, Kim, Thomassé, and Watrigant [8] introduced the twinwidth of a graph. We show that the twin-width of an n-vertex graph is less than pn `?n ln n `?n `2 ln nq{2, and the twin-width of an m-edge graph is less than ? 3m `m1{4 ? ln m{p4 ¨31{4 q `3m 1{4 {2. Conference graphs of order n (when such graphs exist) have twinwidth at least pn ´1q{2, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erdős-Rényi random graph Gpn, pq with 1{n ď p " ppnq ď 1{2 is larger than 2pp1 ´pqn ´p2 ? 2 `εq a pp1 ´pqn ln n asymptotically almost surely for any positive ε. Lastly, we calculate the twin-width of random graphs Gpn, pq with p ď c{n for a constant c ă 1, determining the thresholds at which the twin-width jumps from 0 to 1 and from 1 to 2.

Randomized communication and implicit graph representations

“…Often, graph families where a certain width parameter is bounded possess favorable structural, algorithmic, or combinatorial properties. The powerful twin-width parameter, introduced recently in [15], generalizes treewidth and clique-width and has attracted a lot of recent attention [1,8,25,50,51]. Graph families of bounded twin-width are positive examples to the IGQ [14], making them a natural choice for studying Question 1.2.…”

“…We observed that this would have an unintuitive consequence for communication complexity: computing adjacency in graphs from a hereditary, factorial family would have complexity 𝑂 (1) or Ω(log log 𝑛) but nothing in between; see Example 1.13. This is similar to the jump between 𝑂 (1)-size and Ω(log 𝑛)-size deterministic adjacency labeling schemes, and the jumps in the speed of hereditary graph families, so it seemed reasonable to expect a similar jump for communication.…”

“…But for any non-constant 𝑘 (𝑑), the hereditary graph family 𝔉(HD 𝑘 ) has sketch size Ω(log 𝑛), because it has unbounded chain number. We show that for every 𝑡 ∈ N we can choose 𝑑 sufficiently large to construct disjoint sets {𝑎 (1) , . .…”

“…In this paper we study a new connection between constant-cost randomized 1 communication and a long-standing question about implicit graph representations in structural graph theory [35,52]. Randomized communication is a central topic in communication complexity [17,26,28,43], and the most basic question is, Does a given problem have a randomized protocol with only constant cost?…”

“…3 The results of [30] were obtained before the author became aware of [24]. (1) We study an extensive and diverse list of families F that are important in the literature on the IGQ and resolve Question 1.2 for the set of hereditary subfamilies H ⊆ F . We consider monogenic bipartite families, interval graphs, permutation graphs, families of bounded twin-width (including bounded tree-and clique-width), and many others (see Section 1.3).…”