Linear Layouts of Weakly Triangulated Graphs

Abstract

A graph G = (V,E) is said to be triangulated if it has no chordless cycles of length 4 or more. Such a graph is said to be rigid if, for a valid assignment of edge lengths, it has a unique linear layout and non-rigid otherwise. Damaschke [7] showed how to compute all linear layouts of a triangulated graph, for a valid assignment of lengths to the edges of G. In this paper, we extend this result to weakly triangulated graphs, resolving an open problem. A weakly triangulated graph can be constructively characterized by a peripheral ordering of its edges. The main contribution of this paper is to exploit such an edge order to identify the rigid and non-rigid components of G. We first show that a weakly triangulated graph without articulation points has at most $2^{n_q}$ different linear layouts, where n _q is the number of quadrilaterals (4-cycles) in G. When G has articulation points, the number of linear layouts is at most $2^{n_b - 1 + n_q}$, where n _b is the number of nodes in the block tree of G and n _q is the total number of quadrilaterals over all the blocks. Finally, we propose an algorithm for computing a peripheral edge order of G by exploiting an interesting connection between this problem and the problem of identifying a two-pair in $\overline{G}$. Using an $\mathcal{O}(n\cdot m)$ time solution for the latter problem when G has n vertices and m edges, we propose an $\mathcal{O}(n^2\cdot m)$ time algorithm for computing its peripheral edge order. For sparse graphs, the time-complexity can be improved to $\mathcal{O}(m^2)$, using the concept of handles proposed in [1].