K-chordal Graphs Research Articles

Fuzzy Chordal Graphs and Its Properties

Chordal graphs are those graphs which have chords for each cycle of the length > 3. For large graphs/ networks, generally, the number of chords is less in number than the required number of chords to for a chordal graph. By definition, those graphs are not chordal. The algorithms and properties of chordal graphs do not apply to such cases. Also, the strength of the chords is not measured there. This study introduces a relaxation on such number of chords for the definition of chordal graphs. The notion of a fuzzy strong chord is introduced. After that, fuzzy chordal graphs and related properties are developed. The measure of fuzzy strong chords is proposed. Also, as a generation, fuzzy k-chordal graphs are developed, and isomorphism on a fuzzy chordal graph is defined. At last, area of applications of fuzzy chordal graphs and conclusions with future directions are illustrated.

Characterizing k-chordal unichord-free graphs

The [Formula: see text]-chordal graphs, defined by no induced cycle having length greater than [Formula: see text], generalize the traditional chordal graphs (where [Formula: see text]), which Dirac characterized in 1961 by minimal separators always inducing complete subgraphs. The more recent unichord-free graphs, defined by no cycle having a unique chord, have been characterized by minimal separators always inducing edgeless subgraphs. This sharp contrast of minimal separators motivates a new concept of [Formula: see text]-unichord-free graphs, strengthening the unichord-free graphs (where [Formula: see text]). The class of all unichord-free graphs decomposes into subclasses that are simultaneously [Formula: see text]-chordal and [Formula: see text]-unichord-free, leading to characterizations of some of these subclasses and to open questions about their overall structure as [Formula: see text] and [Formula: see text] increase.

K-Chordal Graphs: From Cops and Robber to Compact Routing via Treewidth

Cops and robber games, introduced by Winkler and Nowakowski (in Discrete Math. 43(2---3), 235---239, 1983) and independently defined by Quilliot (in J. Comb. Theory, Ser. B 38(1), 89---92, 1985), concern a team of cops that must capture a robber moving in a graph. We consider the class of k-chordal graphs, i.e., graphs with no induced (chordless) cycle of length greater than k, k?3. We prove that k?1 cops are always sufficient to capture a robber in k-chordal graphs. This leads us to our main result, a new structural decomposition for a graph class including k-chordal graphs. We present a polynomial-time algorithm that, given a graph G and k?3, either returns an induced cycle larger than k in G, or computes a tree-decomposition of G, each bag of which contains a dominating path with at most k?1 vertices. This allows us to prove that any k-chordal graph with maximum degree Δ has treewidth at most (k?1)(Δ?1)+2, improving the O(Δ(Δ?1)k?3) bound of Bodlaender and Thilikos (Discrete Appl. Math. 79(1---3), 45---61, 1997. Moreover, any graph admitting such a tree-decomposition has small hyperbolicity). As an application, for any n-vertex graph admitting such a tree-decomposition, we propose a compact routing scheme using routing tables, addresses and headers of size O(klogΔ+logn) bits and achieving an additive stretch of O(klogΔ). As far as we know, this is the first routing scheme with O(klogΔ+logn)-routing tables and small additive stretch for k-chordal graphs.

A Dirac-type characterization of [formula omitted]-chordal graphs

A graph is k-chordal if it has no induced cycle of length greater than k. We give a new characterization of k-chordal graphs that generalizes the well-known characterization of chordal graphs by Dirac in terms of simplicial vertices and simplicial orderings.

Distributed computing of efficient routing schemes in generalized chordal graphs

Efficient algorithms for computing routing tables should take advantage of particular properties arising in large scale networks. Two of them are of special interest: low (logarithmic) diameter and high clustering coefficient.High clustering coefficient implies the existence of few large induced cycles. Considering this fact, we propose here a routing scheme that computes short routes in the class of k-chordal graphs, i.e., graphs with no induced cycles of length more than k. In the class of k-chordal graphs, our routing scheme achieves an additive stretch of at most k−1, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus k−1.In order to compute the routing tables of any n-node graph with diameter D we propose a distributed algorithm which uses O(logn)-bit messages and takes O(D) time. The corresponding routing scheme achieves the stretch of k−1 on k-chordal graphs. We then propose a routing scheme that achieves a better additive stretch of 1 in chordal graphs (notice that chordal graphs are 3-chordal graphs). In this case, distributed computation of routing tables takes O(min{ΔD,n}) time, where Δ is the maximum degree of the graph. Our routing schemes use addresses of size logn bits and local memory of size 2(d−1)logn bits per node of degree d.

Compact Routing Schemes for Generalised Chordal Graphs

In this paper, we show how to use the notion of layering-tree introduced in [5], in order to obtain polynomial time constructible routing schemes. We describe efficient routing schemes for two classes of graphs that include the class of chordal graphs. For k-chordal graphs, i.e., graphs containing no induced cycle of length greater than k, the routing scheme uses addresses and local memories of size O(log 2 n) bits per node, and the length of the route between all pairs of vertices never exceeds their distance plus k + 1 (deviation at most k + 1). For tree-length δ graphs, i.e., graphs admitting a tree-decomposition in which the diameter of any bag is at most δ, the routing scheme uses addresses and local memories of size O(δ log 2 n) bits per node, and its deviation is at most 6δ� 2. Observe that for chordal graphs, for which δ = 1 and k = 3, both schemes produce a deviation 4, with addresses and local memories of size O(log 2 n )b its per node.

Treewidth for graphs with small chordality

A graph G is K-chordal, if it does not contain chordless cycles of length larger than k. The chordality lc of a graph G is the minimum k for which G is k-chordal. The degeneracy or the width of a graph is the maximum min-degree of any of its subgraphs. Our results are the following: 1. (1) The problem of treewidth remains NP-complete when restricted to graphs with small maximum degree. 2. (2) An upper bound is given for the treewidth of a graph as a function of its maximum degree and chordality. A consequence of this result is that when maximum degree and chordality are fixed constants, then there is a linear algorithm for treewidth and a polynomial algorithm for pathwidth. 3. (3) For any constant s ⩾ 1, it is shown that any ( s + 2)-chordal graph with bounded width contains an 1 2- separator of size O(n (s − 1) s ) , computable in O(n 3 −( 1 s ) ) time. Our results extent the many applications of the separator theorems in [1, 32, 33] to the class of K-chordal graphs. Several natural classes of graphs have small chordality. Weakly chordal graphs and cocomparability graphs are 4-chordal. We investigate the complexity of treewidth and pathwidth on these classes when an additional degree restriction is used. We present an application of our separator theorem on approximating the maximum independent set on K-chordal graphs with small width.

The Pathwidth and Treewidth of Cographs

It is shown that the pathwidth of a cograph equals its treewidth, and a linear time algorithm to determine the pathwidth of a cograph and build a corresponding path-decomposition is given.