On Hyper-Chordal graphs

Triangulated graphs have many interesting properties (perfection, recognition algorithms, combinatorial optimization algorithms with linear complexity). Hyper-triangulated graphs are those where each induced subgraph has a hyper-simplicial vertex. In this paper we give the characterizations of hyper-triangulated graphs using an ordering of vertices and the weak decomposition. We also offer a recognition algorithm for the hyper-triangulated graphs, the inclusions between the triangulated graphs generalizations and we show that any hyper-triangulated graph is perfect.

Linear layouts of weakly triangulated graphs

A graph [Formula: see text] 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 [Formula: see text]. 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 [Formula: see text]. We first show that a weakly triangulated graph without articulation points has at most [Formula: see text] different linear layouts, where [Formula: see text] is the number of quadrilaterals (4-cycles) in [Formula: see text]. When [Formula: see text] has articulation points, the number of linear layouts is at most [Formula: see text], where [Formula: see text] is the number of nodes in the block tree of [Formula: see text] and [Formula: see text] is the total number of quadrilaterals over all the blocks. Finally, we propose an algorithm for computing a peripheral edge order of [Formula: see text] by exploiting an interesting connection between this problem and the problem of identifying a two-pair in [Formula: see text]. Using an [Formula: see text] time solution for the latter problem, we propose an [Formula: see text] time algorithm for computing its peripheral edge order, where [Formula: see text] and [Formula: see text] are respectively the number of edges and vertices of [Formula: see text]. For sparse graphs, the time complexity can be improved to [Formula: see text], 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].

THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.