A polynomial delay algorithm generating all potential maximal cliques in triconnected planar graphs

TL;DR

This work addresses the problem of enumerating all potential maximal cliques (PMCs) and computing planar treewidth by exploiting planarity. It introduces a new latching-graph framework and a steering-based characterization that ties PMCs in triconnected planar graphs to structured subgraphs, enabling a polynomial-delay generator for PMCs. Consequently, the authors obtain a planar-treewidth algorithm running in time $|Π(G)|\,n^{O(1)}$ for triconnected graphs and extend the approach to general planar graphs via almost-clique handling of two-vertex separators. This yields a practical, planarity-aware exact method for treewidth computations and advances the understanding of PMC enumeration in structured graph classes.

Abstract

We develop a new characterization of potential maximal cliques of a triconnected planar graph and, using this characterization, give a polynomial delay algorithm generating all potential maximal cliques of a given triconnected planar graph. Combined with the dynamic programming algorithms due to Bouchitt{é} and Todinca, this algorithm leads to a treewidth algorithm for general planar graphs that runs in time linear in the number of potential maximal cliques and polynomial in the number of vertices.

Figures (5)

• Figure 1: (A) a triconnected plane graph (solid edges) and its latching graph (solid and broken edges). (B) edge $\{u, v\}$ is drawn in more than one face if $u$ and $v$ separate the graph.
• Figure 2: Some steerings. White vertices belong to $P$ and black vertices belong to $S$, in a choice of the bipartition $(S, P)$ that works. The first three are wheels while the remaining three are non-wheels.
• Figure 3: (A) A steering with three possible $(S, P)$ bipartitions. Because of the first or the second bipartition, it is a wheel. (B) Also a wheel. For the first $(S, P)$ bipartition, it is not an $(S, P)$-steering since an internal vertex of the path on $P$ are adjacent to $s \in S$. The second $(S, P)$ bipartition shows that it is a wheel.
• Figure 4: Some graphs that are not steerings. In the shown $(S, P)$ bipartition, the first example $H$ is not a steering since $N_H(P) \cap S$ is a slot. The second example is not a steering since $N_H(t) \cap S$ is not a slot for the right end $t$ of the path $H[P]$. The third example is not a steering since an internal vertex of the path $H[P]$ is adjacent to a vertex in $S$. It can be verified that these graphs are not $(S, P)$-steerings for any other $(S, P)$ bipartition. It is explained in the main text why any of these plane graphs cannot be $L_G[X]$ for a PMC $X$ of a triconnected plane graph $G$.
• Figure 5: (A) White vertices are some ports of $(S, C)$ where $S$ consists of black vertices and $C$ is the full component associated with $S$ that lies in the outer face of the black cycle. (B) Some valid pairs of ports. The valid pair in the second example has a hinge $s$. (C) Some invalid pairs of ports.

Theorems & Definitions (34)

• Lemma 1: Bouchitté and Todinca bouchitte2001treewidth
• Lemma 2: Bouchitté and Todinca bouchitte2001treewidth
• Theorem 3: Bouchitté and Todinca bouchitte2001treewidth
• Lemma 4: Bouchitté and Todinca bouchitte2001treewidth
• Definition 5
• Proposition 6
• Proposition 7
• Proposition 8
• Proposition 9
• Remark 10