WALCOM 2014: Algorithms and Computation pp 322-336

# Linear Layouts of Weakly Triangulated Graphs

• S. V. Rao
• Sidharth Pardeshi
• Srinivas Gundlapalli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8344)

## 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Hayward, R.B., Spinrad, J.P., Sritharan, R.: Improved algorithms for weakly chordal graphs. ACM Trans. Algorithms 3(2) (2007)Google Scholar
2. 2.
Alam, M.S., Mukhopadhyay, A.: Improved upper and lower bounds for the point placement problem. Technical report, University of Windsor (2010)Google Scholar
3. 3.
Saxe, J.B.: Embeddability of weighted graphs in k-space is strongly NP-hard. In: 17th Allerton Conference on Communication, Control and Computing, pp. 480–489 (1979)Google Scholar
4. 4.
Barvinok, A.I.: Problems of distance geometry and convex propeties of quadratic maps. Discrete and Computational Geometry 13, 189–202 (1995)
5. 5.
Alfakih, A.Y., Wolkowicz, H.: On the embeddability of weighted graphs in euclidean spaces. Technical report, University of Waterloo (1998)Google Scholar
6. 6.
Hastad, J., Ivansson, L., Lagergren, J.: Fitting points on the real line and its application to rh mapping. Journal of Algorithms 49, 42–62 (2003)
7. 7.
Damaschke, P.: Point placement on the line by distance data. Discrete Applied Mathematics 127(1), 53–62 (2003)
8. 8.
Hayward, R.: Generating weakly triangulated graphs. J. Graph Theory 21, 67–70 (1996)
9. 9.
Arikati, S.R., Rangan, C.P.: An efficient algorithm for finding a two-pair, and its applications. Discrete Applied Mathematics 31(1), 71–74 (1991)

© Springer International Publishing Switzerland 2014