Linear layouts of weakly triangulated graphs

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 [Point placement on the line by distance data, Discrete Appl. Math. 127(1) (2003) 53-62] 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(nq) 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(nb-1+nq), 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 G. Using an O(nm) time solution for the latter problem, we propose an O(n(2)m) time algorithm for computing its peripheral edge order, where m and n are respectively the number of edges and vertices of G. For sparse graphs, the time complexity can be improved to O(m(2)), using the concept of handles [R.B. Hayward, J.P. Spinrad and R. Sritharan, Improved algorithms for weakly chordal graphs, ACM Trans. Algorithms 3(2) (2007) 19pp].