Improved bottleneck domination algorithms

W.C.K. Yen introduced BOTTLENECK DOMINATION and BOTTLENECK INDEPENDENT DOMINATION. He presented an -time algorithm to compute a minimum bottleneck dominating set. He also obtained that the BOTTLENECK INDEPENDENT DOMINATING SET problem is NP-complete, even when restricted to planar graphs.We present simple linear time algorithms for the BOTTLENECK DOMINATING SET and the BOTTLENECK TOTAL DOMINATING SET problem. Furthermore, we give polynomial time algorithms (most of them with linear time-complexities) for the BOTTLENECK INDEPENDENT DOMINATING SET problem on the following graph classes: AT-free graphs, chordal graphs, split graphs, permutation graphs, graphs of bounded treewidth, and graphs of clique-width at most k with a given k-expression.     

The parameterized complexity of the induced matching problem

Given a graph G and an integer k≥0, the NP-complete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs, as well as on many restricted graph classes has been studied intensively. However, other than the fact that the problem is W[1]-hard on general graphs, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we provide first-time fixed-parameter tractability results for planar graphs, bounded-degree graphs, graphs with girth at least six, bipartite graphs, line graphs, and graphs of bounded treewidth. In particular, we give a linear-size problem kernel for planar graphs.     

Linearity of grid minors in treewidth with applications through bidimensionality

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minorfree graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Our result has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms. A preliminary version of this paper appeared in the ACM-SIAM Symposium on Discrete Algorithms (SODA 2005) [16].     

Boxicity of Halin graphs

A k-dimensional box is the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K₄, then . In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then unless G is isomorphic to K₄ (in which case its boxicity is 1).     

Degree distribution in random planar graphs

We prove that for each k?0, the probability that a root vertex in a random planar graph has degree k tends to a computable constant dk, so that the expected number of vertices of degree k is asymptotically dkn, and moreover that k_∑dk=1. The proof uses the tools developed by Giménez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=k_∑dkwk. From this we can compute the dk to any degree of accuracy, and derive the asymptotic estimate dk∼c⋅k−1/2qk for large values of k, where q≈0.67 is a constant defined analytically.     

On maximizing clique, clique-Helly and hereditary clique-Helly induced subgraphs

Clique-Helly and hereditary clique-Helly graphs are polynomial-time recognizable. Recently, we presented a proof that the clique graph recognition problem is NP-complete [L. Alcón, L. Faria, C.M.H. de Figueiredo, M. Gutierrez, Clique graph recognition is NP-complete, in: Proc. WG 2006, in: Lecture Notes in Comput. Sci., vol. 4271, Springer, 2006, pp. 269-277]. In this work, we consider the decision problems: given a graph G=(V,E) and an integer k≥0, we ask whether there exists a subset V^′⊆V with |V^′|≥k such that the induced subgraph G[V^′] of G is, variously, a clique, clique-Helly or hereditary clique-Helly graph. The first problem is clearly NP-complete, from the above reference; we prove that the other two decision problems mentioned are NP-complete, even for maximum degree 6 planar graphs. We consider the corresponding maximization problems of finding a maximum induced subgraph that is, respectively, clique, clique-Helly or hereditary clique-Helly. We show that these problems are Max SNP-hard, even for maximum degree 6 graphs. We show a general polynomial-time -approximation algorithm for these problems when restricted to graphs with fixed maximum degree Δ. We generalize these results to other graph classes. We exhibit a polynomial 6-approximation algorithm to minimize the number of vertices to be removed in order to obtain a hereditary clique-Helly subgraph.     

Polar cographs

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is (s,k)-polar if there exists a partition A,B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i.e., the complement of a complete k-partite graph).Recognizing a polar graph is known to be NP-complete. These graphs have not been extensively studied and no good characterization is known. Here we consider the class of polar graphs which are also cographs (graphs without induced path on four vertices). We provide a characterization in terms of forbidden subgraphs. Besides, we give an algorithm in time O(n) for finding a largest induced polar subgraph in cographs; this also serves as a polar cograph recognition algorithm. We examine also the monopolar cographs which are the (s,k)-polar cographs where min(s,k)?1. A characterization of these graphs by forbidden subgraphs is given. Some open questions related to polarity are discussed.     

Rooted directed path graphs are leaf powers

Leaf powers are a graph class which has been introduced to model the problem of reconstructing phylogenetic trees. A graph G=(V,E) is called k-leaf power if it admits a k-leaf root, i.e., a tree T with leaves V such that uv is an edge in G if and only if the distance between u and v in T is at most k. Moroever, a graph is simply called leaf power if it is a k-leaf power for some k∈N. This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a given graph is a k-leaf power.We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a well-known graph class with absolutely different motivations. After this, we provide the largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path graphs.Subsequently, we study the leaf rank problem, the algorithmic challenge of determining the minimum k for which a given graph is a k-leaf power. Firstly, we give a lower bound on the leaf rank of a graph in terms of the complexity of its separators. Secondly, we use this measure to show that the leaf rank is unbounded on both the class of ptolemaic and the class of unit interval graphs. Finally, we provide efficient algorithms to compute 2|V|-leaf roots for given ptolemaic or (unit) interval graphs G=(V,E).     

Four gravity results

The gravity of a graph H in a given family of graphs H is the greatest integer n with the property that for every integer m, there exists a supergraph G∈H of H such that each subgraph of G, which is isomorphic to H, contains at least n vertices of degree ?m in G. Madaras and Škrekovski introduced this concept and showed that the gravity of the path P_k on k?2 vertices in the family of planar graphs of minimum degree 2 is k-2 for each k≠5,7,8,9. They conjectured that for each of the four excluded cases the gravity is k-3. In this paper we show that this holds.     

Contraction obstructions for treewidth

We provide two parameterized graphs Γk, Πk with the following property: for every positive integer k, there is a constant ck such that every graph G with treewidth at least ck, contains one of Kk, Γk, Πk as a contraction, where Kk is a complete graph on k vertices. These three parameterized graphs can be seen as “obstruction patterns” for the treewidth with respect to the contraction partial ordering. We also present some refinements of this result along with their algorithmic consequences.     

Interval Routing onk-Trees

A graph has an optimall-interval routing scheme if it is possible to direct messages along shortest paths by labeling each edge with at mostlpairwise-disjoint subintervals of the cyclic interval [1…n] (where each node of the graph is labeled by an integer in the range). Although much progress has been made forl = 1, there is as yet no general tight characterization of the classes of graphs associated with largerl. Bodlaenderet al. have shown that under the assumption of dynamic cost links, each graph with an optimall-interval routing scheme has treewidth of at most 4l. For the setting without dynamic cost links, this paper addresses the complementary question of the number of intervals required to label classes of graphs of treewidthk. Although it has been shown that there exist graphs of treewidth 2 that require a nonconstant number of intervals, our work demonstrates a class of graphs of treewidth 2, namely 2-trees, that are guaranteed to allow 3-interval routing schemes. In contrast, this paper presents a 2-tree that cannot have a 2-interval routing scheme. For generalk, anyk-tree is shown to have an optimal interval routing scheme using 2^{k + 1}intervals per edge.     

Improved induced matchings in sparse graphs

An induced matching in a graph G=(V,E) is a matching M such that (V,M) is an induced subgraph of G. Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj et al. [10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least and that there are twinless planar graphs that do not contain an induced matching of size greater than . We improve both these bounds to , which is tight up to an additive constant. This implies that the problem of deciding whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is fixed parameter tractable for graphs of bounded arboricity.Kanj et al. also presented an algorithm which decides in -time whether an n-vertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we also show a more efficient -time algorithm. Its main ingredient is a new, O^∗(4^l)-time algorithm for finding a maximum induced matching in a graph of branch width at most l.     

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P₈-free bipartite graphs, but polynomial for P₅-free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6−ε, for any ε>0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.     

Treewidth of Cartesian Products of Highly Connected Graphs

The following theorem is proved: for all k‐connected graphs G and H each with at least n vertices, the treewidth of the cartesian product of G and H is at least . For , this lower bound is asymptotically tight for particular graphs G and H. This theorem generalizes a well‐known result about the treewidth of planar grid graphs.     

Perfect matchings after vertex deletions

This paper considers some classes of graphs which are easily seen to have many perfect matchings. Such graphs can be considered robust with respect to the property of having a perfect matching if under vertex deletions (with some mild restrictions), the resulting subgraph continues to have a perfect matching. It is clear that you can destroy the property of having a perfect matching by deleting an odd number of vertices, by upsetting a bipartition or by deleting enough vertices to create an odd component. One class of graphs we consider is the m×m lattice graph (or grid graph) for m even. Matchings in such grid graphs correspond to coverings of an m×m checkerboard by dominoes. If in addition to the easy conditions above, we require that the deleted vertices be apart, the resulting graph has a perfect matching. The second class of graphs we consider is a k-fold product graph consisting of k copies of a given graph G with the ith copy joined to the i+1st copy by a perfect matching joining copies of the same vertex. We show that, apart from some easy restrictions, we can delete any vertices from the kth copy of G and find a perfect matching in the product graph with k suitably large.     

Edge intersection graphs of systems of paths on a grid with a bounded number of bends

We answer some of the questions raised by Golumbic, Lipshteyn and Stern and obtain some other results about edge intersection graphs of paths on a grid (EPG graphs). We show that for any d≥4, in order to represent every n vertex graph with maximum degree d as an edge intersection graph of n paths on a grid, a grid of area Θ(n²) is needed. We also show several results related to the classes B_k-EPG, where B_k-EPG denotes the class of graphs that have an EPG representation such that each path has at most k bends. In particular, we prove: For a fixed k and a sufficiently large n, the complete bipartite graph K_m,n does not belong to B_2m−3-EPG (it is known that this graph belongs to B_2m−2-EPG); for any odd integer k we have B_k-EPG B_k+1-EPG; there is no number k such that all graphs belong to B_k-EPG; only 2^O(knlog(kn)) out of all the labeled graphs with n vertices are in B_k-EPG.     

On the maximum cardinality search lower bound for treewidth

The Maximum Cardinality Search (MCS) algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbours becomes visited. A maximum cardinality search ordering (MCS-ordering) of a graph is an ordering of the vertices that can be generated by the MCS algorithm. The visited degree of a vertex v in an MCS-ordering is the number of neighbours of v that are before v in the ordering. The visited degree of an MCS-ordering ψ of G is the maximum visited degree over all vertices v in ψ. The maximum visited degree over all MCS-orderings of graph G is called its maximum visited degree. Lucena [A new lower bound for tree-width using maximum cardinality search, SIAM J. Discrete Math. 16 (2003) 345-353] showed that the treewidth of a graph G is at least its maximum visited degree.We show that the maximum visited degree is of size O(logn) for planar graphs, and give examples of planar graphs G with maximum visited degree k with O(k!) vertices, for all k∈N. Given a graph G, it is NP-complete to determine if its maximum visited degree is at least k, for any fixed k?7. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete.In this paper, we also propose some heuristics for the problem, and report on an experimental analysis of them. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.     

Mutual exclusion scheduling with interval graphs or related classes, Part I

In this paper, the mutual exclusion scheduling problem is addressed. Given a simple and undirected graph G and an integer k, the problem is to find a minimum coloring of G such that each color is used at most k times. When restricted to interval graphs or related classes like circular-arc graphs and tolerance graphs, the problem has some applications in workforce planning. Unfortunately, the problem is shown to be NP-hard for interval graphs, even if k is a constant greater than or equal to four [H.L. Bodlaender and K. Jansen Restrictions of graph partition problems. Part I, Theoretical Computer Science 148(1995) pp. 93-109]. Several polynomial-time solvable cases significant in practice are exhibited here, for which we took care to devise simple and efficient algorithms (in particular linear-time and space algorithms). On the other hand, by reinforcing the NP-hardness result of Bodlaender and Jansen, we obtain a more precise cartography of the complexity of the problem for the classes of graphs studied.     

Graph parameters measuring neighbourhoods in graphs—Bounds and applications

The neighbourhood-width of a graph G=(V,E) is introduced in [F. Gurski, Linear layouts measuring neighbourhoods in graphs, Discrete Math. 306 (15) (2006) 1637-1650.] as the smallest integer k such that there is a linear layout ?:V→{1,…,|V|} such that for every 1?i<|V| the vertices u with ?(u)?i can be divided into at most k subsets each members having the same neighbours with respect to the vertices v with ?(v)>i.In this paper we show first bounds for the neighbourhood-width of general graphs, caterpillars, trees and grid graphs and give applications of the layout parameter neighbourhood-width in graph drawing and VLSI design.