Showing papers by "Pinar Heggernes published in 2009"
TL;DR: A new representation of proper interval graphs that can be computed in linear time and stored in O ( n ) space is introduced, a 2-dimensional vertex partition that is particularly interesting with respect to clique-width.
30 citations
TL;DR: It is proved that split graphs have the following property: given two split graphs on the same vertex set where one is a subgraph of the other, there is a sequence of edges that can be removed from the larger to obtain the smaller such that after each edge removal the modified graph is split.
23 citations
TL;DR: This work gives the first polynomial-time algorithm that computes the bandwidth of bipartite permutation graphs, which is an NP-complete graph layout problem that is notorious for its difficulty even on small graph classes.
23 citations
TL;DR: It is proved that threshold graphs and chain graphs admit such sequences, and linear-time algorithms are presented both for computing minimal completions and deletions into threshold, chain, and bipartite graphs, and for extracting a minimal completion or delete from a given completion or deletion.
17 citations
10 Nov 2009
TL;DR: It is shown that for permutation graphs this problem can be solved in polynomial time, which is surprising, as related problems like achromatic number and co chromatic number are NP-complete onpermutation graphs.
Abstract: Polar graphs generalise bipartite, cobipartite, split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs.
17 citations
TL;DR: An algorithm is presented that supports operations for modifying asplit graph by adding edges or vertices and deleting edges, such that after each modification the graph is repaired to become a split graph in a minimal way.
16 citations
20 Aug 2009
TL;DR: It is proved that the problem is fixed parameter tractable on P 5-free graphs when parameterized by k, which contains the well known and widely studied class of cographs.
Abstract: A graph is k-choosable if it admits a proper coloring of its vertices for every assignment of k (possibly different) allowed colors to choose from for each vertex. It is NP-hard to decide whether a given graph is k-choosable for k ? 3, and this problem is considered strictly harder than the k-coloring problem. Only few positive results are known on input graphs with a given structure. Here, we prove that the problem is fixed parameter tractable on P 5-free graphs when parameterized by k. This graph class contains the well known and widely studied class of cographs. Our result is surprising since the parameterized complexity of k-coloring is still open on P 5-free graphs. To give a complete picture, we show that the problem remains NP-hard on P 5-free graphs when k is a part of the input.
13 citations
12 May 2009
TL;DR: It is shown that k -path powers above a certain size have linear clique-width exactly k + 2, providing the first complete characterisation of the linear cliques-width of a graph class of unbounded clique -width.
Abstract: A k -path power is the k -power graph of a simple path of arbitrary length. Path powers form a non-trivial subclass of proper interval graphs. Their clique-width is not bounded by a constant, and no polynomial-time algorithm is known for computing their clique-width or linear clique-width. We show that k -path powers above a certain size have linear clique-width exactly k + 2, providing the first complete characterisation of the linear clique-width of a graph class of unbounded clique-width. Our characterisation results in a simple linear-time algorithm for computing the linear clique-width of all path powers.
10 citations
05 Dec 2009
TL;DR: The result is the first non-trivial FPT algorithm for Bandwidth on a graph class where the problem remains NP-complete, and on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs.
Abstract: We study the classical Bandwidth problem from the viewpoint of parameterized algorithms. In the Bandwidth problem we are given a graph G = (V,E) together with a positive integer k, and asked whether there is an bijective function β: {1, ..., n} ?V such that for every edge uv ? E, |β ? 1(u) ? β ? 1(v)| ≤ k. The problem is notoriously hard, and it is known to be NP-complete even on very restricted subclasses of trees. The best known algorithm for Bandwidth for small values of k is the celebrated algorithm by Saxe [SIAM Journal on Algebraic and Discrete Methods, 1980 ], which runs in time $2^{{\mathcal{O}}(k)}n^{k+1}$. In a seminal paper, Bodlaender, Fellows and Hallet [STOC 1994 ] ruled out the existence of an algorithm with running time of the form $f(k)n^{{\mathcal{O}}(1)}$ for any function f even for trees, unless the entire W-hierarchy collapses.
We initiate the search for classes of graphs where Bandwidth is fixed parameter tractable (FPT), that is, solvable in time $f(k)n^{{\mathcal{O}}(1)}$ for some function f. In this paper we present an algorithm with running time $2^{{\mathcal O}(k \log k)} n^2$ for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial FPT algorithm for Bandwidth on a graph class where the problem remains NP-complete.
9 citations
20 Jun 2009
TL;DR: A linear-time algorithm is given for computing the edge search number of cographs, thereby proving that this problem can be solved in polynomial time on this graph class.
Abstract: We give a linear-time algorithm for computing the edge search number of cographs, thereby proving that this problem can be solved in polynomial time on this graph class. With our result, the knowledge on graph searching of cographs is now complete: node, mixed, and edge search numbers of cographs can all be computed efficiently. Furthermore, we are one step closer to computing the edge search number of permutation graphs.
7 citations
11 Jul 2009
TL;DR: This paper shows that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono from 1997 and following such algorithms as polynomial-time algorithms for both classes.
Abstract: A graph class is sandwich monotone if, for every pair of its graphs G 1 = (V ,E 1 ) and G 2 = (V ,E 2 ) with E 1 *** E 2 , there is an ordering e 1 , ..., e k of the edges in E 2 *** E 1 such that G = (V , E 1 *** {e 1 , ..., e i }) belongs to the class for every i between 1 and k . In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono from 1997. So far, very few classes have been proved to be sandwich monotone, and the most famous of these are chordal graphs. Sandwich monotonicity of a graph class implies that minimal completions of arbitrary graphs into that class can be recognized and computed in polynomial time. For minimal completions into strongly chordal or chordal bipartite graphs no polynomial-time algorithm has been known. With our results such algorithms follow for both classes. In addition, from our results it follows that all strongly chordal graphs and all chordal bipartite graphs with edge constraints can be listed efficiently.