Weakly triangulated graphs

A graph is triangulated if it has no chordless cycle with four or more vertices. It follows that the complement of a triangulated graph cannot contain a chordless cycle with five or more vertices. We introduce a class of graphs (namely, weakly triangulated graphs) which includes both triangulated graphs and complements of triangulated graphs (we define a graph as weakly triangulated if neither it nor its complement contains a chordless cycle with five or more vertices). Our main result is a structural theorem which leads to a proof that weakly triangulated graphs are perfect.     

Chromaticity of triangulated graphs

It is shown here that a connected graph G without subgraphs isomorphic to K₄ is triangulated if and only if its chromatic polynomial P(G,λ) equals λ(λ ? 1)^m(λ ? 2)^r for some integers m ≧ 1, r ≧ 0. This result generalizes the characterization of Two-Trees given by E.G. Whitehead [“Chromaticity of Two-Trees,” Journal of Graph Theory 9 (1985) 279–284].     

Optimizing weakly triangulated graphs

A graph is weakly triangulated if neither the graph nor its complement contains a chordless cycle with five or more vertices as an induced subgraph. We use a new characterization of weakly triangulated graphs to solve certain optimization problems for these graphs. Specifically, an algorithm which runs inO((n + e)n ³) time is presented which solves the maximum clique and minimum colouring problems for weakly triangulated graphs; performing the algorithm on the complement gives a solution to the maximum stable set and minimum clique covering problems. Also, anO((n + e)n ⁴) time algorithm is presented which solves the weighted versions of these problems.The author acknowledges the support of an N.S.E.R.C. Canada postgraduate scholarship.The author acknowledges the support of the U.S. Air Force Office of Scientific Research under grant number AFOSR 0271 to Rutgers University.     

Generating weakly triangulated graphs

We show that a graph is weakly triangulated, or weakly chordal, if and only if it can be generated by starting with a graph with no edges, and repeatedly adding an edge, so that the new edge is not the middle edge of any chordless path with four vertices. This is a corollary of results due to Sritharan and Spinrad, and Hayward, Hoång and Maffray, and a natural analog of a theorem due to Fulkerson and Gross, which states that a graph is triangulated, or chordal, if and only if it can be generated by starting with a graph with no vertices, and repeatedly adding a vertex, so that the new vertex is not the middle vertex of any chordless path with three vertices. Our result answers the question of whether there exists a composition scheme that generates exactly the class of weakly triangulated graphs. © 1996 John Wiley & Sons, Inc.     

More characterizations of triangulated graphs

New characterizations of triangulated and cotriangulated graphs are presented. Cotriangulated graphs form a natural subclass of the class of strongly perfect graphs, and they are also characterized in terms of the shellability of some associated collection of sets. Finally, the notion of stability function of a graph is introduced, and it is proved that a graph is triangulated if and only if the polynomial representing its stability function has all its coefficients equal to 0, +1 or ?1.     

Slightly triangulated graphs are perfect

A graph istriangulated if it has no chordless cycle with at least four vertices (?k ≥ 4,C _k ?G). These graphs Jhave been generalized by R. Hayward with theweakly triangulated graphs $(\forall k \geqslant 5,C_{k,} \bar C_k \nsubseteq G)$ . In this note we propose a new generalization of triangulated graphs. A graph G isslightly triangulated if it satisfies the two following conditions;

1. G contains no chordless cycle with at least 5 vertices.
2. For every induced subgraphH of G, there is a vertex inH the neighbourhood of which inH contains no chordless path of 4 vertices.

     

GLOUDS: Representing tree-like graphs

The Graph Level Order Unary Degree Sequence (GLOUDS) is a new succinct data structure for directed graphs that are “tree-like,” in the sense that the number of “additional” edges (w.r.t. a spanning tree) is not too high. The algorithmic idea is to represent a BFS-spanning tree of the graph (consisting of n nodes) with a well known succinct data structure for trees, named LOUDS, and enhance it with additional information that accounts for the non-tree edges. In practical tests, our data structure performs well for graphs containing up to $m=5n$ edges, while still having competitive running times for listing adjacent nodes.     

A relationship between triangulated graphs,comparability graphs,proper interval graphs,proper circular-arc graphs,and nested interval graphs

Given a set F of digraphs, we say a graph G is a F-graph (resp., F*-graph) if it has an orientation (resp., acyclic orientation) that has no induced subdigraphs isomorphic to any of the digraphs in F. It is proved that all the classes of graphs mentioned in the title are F-graphs or F*-graphs for subsets F of a set of three digraphs.     

Geometric graphs which are 1-skeletons of unstacked triangulated polygons

We give sufficient (and necessary) conditions of local character ensuring that a geometric graph is the 1-skeleton of an unstacked triangulation of a simple polygon.     

Path problems in generalized stars, complete graphs, and brick wall graphs

Path problems such as the maximum edge-disjoint paths problem, the path coloring problem, and the maximum path coloring problem are relevant for resource allocation in communication networks, in particular all-optical networks. In this paper, it is shown that maximum path coloring can be solved optimally in polynomial time for bidirected generalized stars, even in the weighted case. Furthermore, the maximum edge-disjoint paths problem is proved NP-hard for complete graphs (undirected or bidirected), a constant-factor off-line approximation algorithm is presented for the weighted case, and an on-line algorithm with constant competitive ratio is given for the unweighted case. Finally, an open problem concerning the existence of routings that simultaneously minimize the maximum load and the number of colors is solved: an example for a graph and a set of requests is given such that any routing that minimizes the maximum load requires strictly more colors for path coloring than a routing that minimizes the number of colors.     

Hypergraphs with cyclomatic number zero,triangulated graphs,and an inequality

A parameter μ for hypergraphs generalizing the cyclomatic number of graphs is defined. Theorem: a hypergraph has μ = 0 if and only if its edges maximal with respect to inclusion are the cliques of a triangulated graph. The equality μ = 0 immediately implies an inequality previously proved by L. Lovász and by P. Hansen and M. Las Vergnas for more restricted classes of hypergraphs.     

Representing products of polyhedra by products of edge-colored graphs

Given an (m + 1)-colored graph ( Γ γ′) and an (n + 1)-colored graph (, γ″), representing two polyhedra P′, P″ respectively, we present a direct construction of an (m + n+1)-colored graph ( Γ ′? Γ ″, γ′ ? γ″), which represents the product P ′ × P ″. Some examples, applications, and conjectures about the genus of manifold products are also presented. © 1993 John Wiley & Sons, Inc.     

Representing groups by graphs with constant link and hypergraphs

A graph L is called a link graph if there is a graph G such that for each vertex of G its neighbors induce a subgraph isomorphic to L. Such a G is said to have constant link L. We prove that for any finite group Γ and any disconnected link graph L with at least three vertices there are infinitely many connected graphs G with constant link L and AutG ? Γ. We look at the analogous problem for connected link graphs, namely, link graphs that are paths or have disconnected complements. Furthermore we prove that for n, r ≥ 2, but not n = 2 = r, any finite group can be represented by infinitely many connected r-uniform, n-regular hypergraphs of arbitrarily large girth.     

Connectivity keeping stars or double-stars in 2-connected graphs

In Mader (2010), Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta \left(G\right)\ge ?\frac{3}{2}k?+m?1$ contains a subtree $T^{\prime }$ isomorphic to $T$ such that $G?V\left(T^{\prime }\right)$ is $k$-connected. In the same paper, Mader proved that the conjecture is true when $T$ is a path. Diwan and Tholiya (2009) verified the conjecture when $k=1$. In this paper, we will prove that Mader’s conjecture is true when $T$ is a star or double-star and $k=2$.