# 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].

## Keywords

Articulation Point Rigid Component Complement Graph Biconnected Component Chordless Cycle## Preview

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