Chordal Bipartite Graphs Research Articles

Relationships between the class of unit grid intersection graphs and other classes of bipartite graphs

We show that the class of unit grid intersection graphs properly includes both of the classes of interval bigraphs and of P 6 -free chordal bipartite graphs. We also demonstrate that the classes of unit grid intersection graphs and of chordal bipartite graphs are incomparable.

Chordal bipartite completion of colored graphs

Golumbic, Kaplan, and Shamir [Graph sandwich problems, J. Algorithms 19 (1995) 449–473], in their paper on graph sandwich problems published in 1995, left the status of the sandwich problems for strongly chordal graphs and chordal bipartite graphs open. It was recently shown [C.M.H. de Figueiredo, L. Faria, S. Klein, R. Sritharan, On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs, Theoret. Comput. Sci., accepted for publication] that the sandwich problem for strongly chordal graphs is NP-complete. We show that given graph G with a proper vertex coloring c, determining whether there is a supergraph of G that is chordal bipartite and also is properly colored by c is NP-complete. This implies that the sandwich problem for chordal bipartite graphs is also NP-complete.

On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs

Golumbic, Kaplan, and Shamir, in their paper [M.C. Golumbic, H. Kaplan, R. Shamir, Graph sandwich problems, J. Algorithms 19 (1995) 449–473] on graph sandwich problems published in 1995, left the status of sandwich problems for strongly chordal graphs and chordal bipartite graphs open. We prove that the sandwich problem for strongly chordal graphs is NP-complete. We also give some comments on the computational complexity of the sandwich problem for chordal bipartite graphs.

Train Tracks and Confluent Drawings

Confluent graphs capture the connection properties of train tracks, offering a very natural generalization of planar graphs, and—as the example of railroad maps shows—are an important tool in graph visualization. In this paper we continue the study of confluent graphs, introducing strongly confluent graphs and tree-confluent graphs. We show that strongly confluent graphs can be recognized in NP (the complexity of recognizing confluent graphs remains open). We also give a natural elimination ordering characterization of tree-confluent graphs, and we show that this class coincides with the (6,2)-chordal bipartite graphs. Finally, we define outerconfluent graphs and identify the bipartite permutation graphs as a natural subclass.

Chordal multipartite graphs and chordal colorings

‘Chordal multipartite graphs’ are properly colored graphs such that two vertices in a minimal vertex separator are adjacent if and only if they are differently colored. They have induced cycle characterizations that transcend those of chordal and chordal bipartite graphs. Graphs that have such ‘chordal colorings’ are weakly chordal graphs with simple forbidden subgraph characterizations, and such a chordal coloring of a graph G requires only χ ( G ) colors.

Graph classes related to chordal graphs and chordal bipartite graphs

Graph classes related to chordal graphs and chordal bipartite graphs

Tree Spanners for Bipartite Graphs and Probe Interval Graphs

A tree t-spanner T in a graph G is a spanning tree of G such that the distance between every pair of vertices in T is at most t times their distance in G. The tree t-spanner problem asks whether a graph admits a tree t-spanner, given t. We first substantially strengthen the known results for bipartite graphs. We prove that the tree t-spanner problem is NP-complete even for chordal bipartite graphs for t ≥ 5, and every bipartite ATE-free graph has a tree 3-spanner, which can be found in linear time. The previous best known results were NP-completeness for general bipartite graphs, and that every convex graph has a tree 3-spanner. We next focus on the tree t-spanner problem for probe interval graphs and related graph classes. The graph classes were introduced to deal with the physical mapping of DNA. From a graph theoretical point of view, the classes are natural generalizations of interval graphs. We show that these classes are tree 7-spanner admissible, and a tree 7-spanner can be constructed in (m log n) time.

Representation characterizations of chordal bipartite graphs

A bipartite graph is chordal bipartite if it does not contain an induced cycle of length at least six. We give three representation characterizations of chordal bipartite graphs. More precisely, we show that a bipartite graph is chordal bipartite if and only if the complement is the intersection graph of a family of pairwise compatible claws in a weighted hypercircle. (A hypercircle is a graph which consists of internally vertex disjoint paths between two distinguished vertices, and a claw in a hypercircle is a connected subgraph containing exactly one of the two distinguished vertices.) We also introduce two classes of bipartite graphs, both containing interval bigraphs and interval containment bigraphs. They are compatible subtree intersection bigraphs and compatible subtree containment bigraphs. We show that these two classes are identical to the class of chordal bipartite graphs.

Collective tree spanners of graphs

In this paper we introduce a new notion of collective tree spanners. We say that a graph G=(V,E)admits a system of $\mu$ collective additive tree r-spanners if there is a system T(G) of at most $\mu$ spanning trees of G such that for any two vertices x,y of G a spanning tree T\in \cT(G) exists such that d_T(x,y)\leq d_G(x,y)+r. Among other results, we show that any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log2n collective additive tree 2-spanners. These results are complemented by lower bounds, which say that any system of collective additive tree 1-spanners must have $\Omega(\sqrt{n})$ spanning trees for some chordal graphs and $\Omega(n)$ spanning trees for some chordal bipartite graphs and some cocomparability graphs. Furthermore, we show that any c-chordal graph admits a system of at most log2n collective additive tree (2\lfloor c/2\rfloor)-spanners, any circular-arc graph admits a system of two collective additive tree 2-spanners. Towards establishing these results, we present a general property for graphs, called (\al,r)$-decomposition, and show that any $(\al,r)$-decomposable graph G with n vertices admits a system of at most $\log_{1/\al} n$ collective additive tree $2r$-spanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs. For any graph on n vertices admitting a system of at most $\mu$ collective additive tree r-spanners, there is a routing scheme of deviation r with addresses and routing tables of size $O(\mu \log^2n/\log \log n)$ bits per vertex. This leads, for example, to a routing scheme of deviation $(2\lfloor c/2\rfloor)$ with addresses and routing tables of size $O(\log^3n/\log \log n)$ bits per vertex on the class of c-chordal graphs.

Estimating all pairs shortest paths in restricted graph families: a unified approach

In this paper we show that a very simple and efficient approach can be used to solve the all pairs almost shortest path problem on the class of weakly chordal graphs and its different subclasses. Moreover, this approach works well also on graphs with small size of largest induced cycle and gives a unified way to solve the all pairs almost shortest path and all pairs shortest path problems on different graph classes including chordal, strongly chordal, chordal bipartite, and distance-hereditary graphs.

Biclique comparability digraphs of bipartite graphs and minimum ranks of partial matrices

Every bipartite graph has a biclique comparability digraph whose vertices are the inclusion-maximal complete bipartite subgraphs of the bipartite graph and whose arcs correspond to inclusions of the relevant color classes. I characterize those digraphs that correspond to bipartite graphs and, in particular, those that correspond to chordal bipartite graphs. This is motivated by work on finding the minimum rank of completions of partially specified matrices. In particular, Woerdeman (Integral Equations Operator Theory 10 (1987) 859) proved a formula for minimum rank in special cases that can be naturally reformulated in terms of the biclique comparability digraphs of the bipartite graphs that have the partial matrices as incidence matrices. Cohen et al. (Oper. Theory Adv. Appl. 40 (1989) 165) conjecture that this formula actually gives the minimum rank if and only if the corresponding bipartite graph is chordal bipartite.

Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs

This paper deals with the graph isomorphism (GI) problem for two graph classes: chordal bipartite graphs and strongly chordal graphs. It is known that GI problem is GI complete even for some special graph classes including regular graphs, bipartite graphs, chordal graphs, comparability graphs, split graphs, and k-trees with unbounded k. On the other hand, the relative complexity of the GI problem for the above classes was unknown. We prove that deciding isomorphism of the classes are GI complete.

Chordal bipartite graphs of bounded tree- and clique-width*1

A bipartite graph is chordal bipartite if every cycle of length at least six has a chord. In the class of chordal bipartite graphs the tree-width and the clique-width are unbounded. Our main results are that chordal bipartite graphs of bounded vertex degree have bounded tree-width and that k -fork-free chordal bipartite graphs have bounded clique-width, where a k -fork is the graph arising from a K 1, k +1 by subdividing one edge once. (Note that a bipartite graph has vertex degree at most k if and only if it is K 1, k +1 -free.) This implies polynomial-time solvability for a variety of algorithmical problems for these graphs.

Chordal bipartite graphs of bounded tree- and clique-width

A bipartite graph is chordal bipartite if every cycle of length at least six has a chord. In the class of chordal bipartite graphs the tree-width and the clique-width are unbounded. Our main results are that chordal bipartite graphs of bounded vertex degree have bounded tree-width and that k-fork-free chordal bipartite graphs have bounded clique-width, where a k-fork is the graph arising from a K 1, k+1 by subdividing one edge once. (Note that a bipartite graph has vertex degree at most k if and only if it is K 1, k+1 -free.) This implies polynomial-time solvability for a variety of algorithmical problems for these graphs.

The bottleneck independent domination on the classes of bipartite graphs and block graphs

Let G( V, E, W) denote a graph with vertex-set V and edge-set E, and each vertex v is associated with a cost W( v). For any set V ′⊆ V, the bottleneck cost of V ′ is defined as max{ W( x)| x∈ V ′}. This paper considers the bottleneck independent dominating set problem (the BIDS problem) which determines an independent dominating set of G such that its bottleneck cost is minimized. This paper studies the problem on the classes of bipartite graphs and block graphs. This paper first proves that the problem is NP-hard on chordal-bipartite graphs. Second, a linear-time algorithm on convex-bipartite graphs is proposed. Next, a linear-time algorithm on trees is designed. Then, we generalize the algorithmic result to block graphs. All algorithms are designed by the dynamic programming strategy.

Subgraph trees in graph theory

The classical clique tree approach to chordal graphs (and, more recently, to strongly chordal graphs) can be generalized to show a common structure for other classes of graphs, including clique graphs of chordal graphs, outerplanar graphs, distance-hereditary graphs, and chordal bipartite graphs.

The Neighborhood Characteristic Parameter for Graphs

Define the neighborhood characteristic of a graph to be $s_1 - s_2 + s_3 - \cdots$, where $s_i$ counts subsets of $i$ vertices that are all adjacent to some vertex outside the subset. This amounts to replacing cliques by neighborhoods in the traditional 'Euler characteristic' (the number of vertices, minus the number of edges, plus the number of triangles, etc.). The neighborhood characteristic can also be calculated by knowing, for all $i,j \ge 2$, how many $K_{i,j}$ subgraphs there are or, through an Euler-Poincaré-type theorem, by knowing how those subgraphs are arranged. Chordal bipartite graphs are precisely the graphs for which every nontrivial connected induced subgraph has neighborhood characteristic 2.

Finding a maximum induced matching in weakly chordal graphs

An induced matching in a graph G is a set of edges, no two of which meet a common vertex or are joined by an edge of G; that is, an induced matching is a matching which forms an induced subgraph. It is known that finding an induced matching of maximum cardinality in a graph is NP-hard. We show that a maximum induced matching in a weakly chordal graph can be found in polynomial time. This generalizes previously known results for the induced matching problem. This also demonstrates that the maximum induced matching problem in chordal bipartite graphs can be solved in polynomial time while the problem is known to be NP-hard for bipartite graphs in general.

Efficient minus and signed domination in graphs

An efficient minus (respectively, signed) dominating function of a graph G=(V,E) is a function f:V→{−1,0,1} (respectively, {−1,1}) such that ∑u∈N[v]f(u)=1 for all v∈V, where N[v]={v}∪{u|(u,v)∈E}. The efficient minus (respectively, signed) domination problem is to find an efficient minus (respectively, signed) dominating function of G. In this paper, we show that the efficient minus (respectively, signed) domination problem is NP-complete on chordal graphs, chordal bipartite graphs, planar bipartite graphs and planar graphs of maximum degree 4 (respectively, on chordal graphs). Based on the forcing property on blocks of vertices and automata theory, we provide a uniform approach to show that in a special class of interval graphs, every graph (respectively, every graph with no vertex of odd degree) has an efficient minus (respectively, signed) dominating function. We also give linear-time algorithms to find these functions. Besides, we show that the efficient minus domination problem is equivalent to the efficient domination problem on trees.

On the k-path partition of graphs

The k-path partition problem is to partition a graph into the minimum number of paths, so that none of them has length more than k, for a given positive integer k. The problem is a generalization of the Hamiltonian path problem and the problem of partitioning a graph into the minimum number of paths. The k-path partition problem remains NP-complete on the class of chordal bipartite graphs if k is part of the input, and we show that it is NP-complete on the class of comparability graphs even for k=3. On the positive side, we present a polynomial-time solution for the problem, with any k, on bipartite permutation graphs, which form a subclass of chordal bipartite graphs.