Haiko Müller

Hamiltonian circuits in chordal bipartite graphs

Abstract The main result of this paper is the NP-completeness of the HAMILTONIAN CIRCUIT problem for chordal bipartite graphs. This is proved by a sophisticated reduction from SATISFIABILITY. As a corollary, HAMILTONIAN CIRCUIT is NP-complete for strongly chordal split graphs. On both classes the complexity of the HAMILTONIAN PATH problem coincides with the complexity of HAMILTONIAN CIRCUIT. Further, we show that HAMILTONIAN CIRCUIT is linear-time solvable for convex bipartite graphs.

Recognizing interval digraphs and interval bigraphs in polynomial time

Abstract An interval bigraph is an undirected bipartite graph whose edge set is the intersection of the edge sets of an interval graph and the edge set of a complete bipartite graph on the same vertex set. A bipartite interval representation of an interval bigraph is given by a bipartitioned set of intervals for its vertices, such that vertices are adjacent if and only if the corresponding intervals intersect and belong to opposite sides of the bipartition. Interval digraphs are directed graphs defined by a closely related concept. Each vertex of an interval digraph is represented by two intervals on the real line, a source interval and a target interval. The directed arc ( u , v ) exists in the interval digraph if the source interval of u meets the target interval of v . We give a dynamic programming algorithm recognizing interval bigraphs (interval digraphs) in polynomial time. This algorithm recursively constructs a bipartite interval representation of a graph from bipartite interval representations of proper subgraphs. Moreover, we list some forbidden substructures of interval bigraphs.

Finding and counting small induced subgraphs efficiently

Abstract We give two algorithms for listing all simplicial vertices of a graph running in time O (n α ) and O (e 2α/(α+1) )= O (e 1.41 ) , respectively, where n and e denote the number of vertices and edges in the graph and O (n α ) is the time needed to perform a fast matrix multiplication. We present new algorithms for the recognition of diamond-free graphs ( O (n α +e 3/2 ) ), claw-free graphs ( O (e (α+1)/2 )= O (e 1.69 ) ), and K 4 -free graphs ( O (e (α+1)/2 )= O (e 1.69 ) ). Furthermore, we show that counting the number of K 4 s in a graph can be done in time O (e (α+1)/2 ) . For all other graphs on four vertices we can count within O (n α +e 1.69 ) time the number of occurrences as induced subgraph.

Domination in convex and chordal bipartite graphs

Abstract We state that various domination problems are polynomial time solvable in convex bipartite graphs, and we give the main ideas of the algorithms. TOTAL DOMINATING SET is polynomial even for chordal bipartite graphs. Further, it is shown by a reduction from 3SAT that INDEPENDENT DOMINATING SET remains NP-complete when restricted to chordal bipartite graphs.

The NP-completeness of steiner tree and dominating set for chordal bipartite graphs

We show that the problems steiner tree, dominating set and connected dominating set are NP-complete for chordal bipartite graphs.

Approximating the Bandwidth for Asteroidal Triple-Free Graphs

We show that there is an O(nm) algorithm to approximate the bandwidth of an AT-free graph with worst case performance ratio 2. Alternatively, at the cost of the approximation factor, we can also obtain an O(m+nlogn) algorithm to approximate the bandwidth of an AT-free graph within a factor 4. For the special cases of permutation graphs and trapezoid graphs we obtain O(nlog2n) algorithms with worst-case performance ratio 2. For cocomparability graphs we obtain an O(n+m) algorithm with worst-case performance ratio 3.

On Vertex Ranking for Permutations and Other Graphs

In this paper we show that an optimal vertex ranking of a permutation graph can be computed in time O(n6), where n is the number of vertices. The demonstrated minimal separator approach can also be used for designing polynomial time algorithms computing an optimal vertex ranking on the following classes of well-structured graphs: circular permutation graphs, interval graphs, circular arc graphs, trapezoid graphs and cocomparability graphs of bounded dimension.

On treewidth approximations

We introduce a natural heuristic for approximating the treewidth of graphs. We prove that this heuristic gives a constant factor approximation for the treewidth of graphs with bounded asteroidal number. Using a different technique, we give a O(log k) approximation algorithm for the treewidth of arbitrary graphs, where k is the treewidth of the input graph.

Computing Treewidth and Minimum Fill-In: All You Need are the Minimal Separators

Consider a class of graphs \(\mathcal{G}\) having a polynomial time algorithm computing the set of all minimal separators for every graph in \(\mathcal{G}\). We show that there is a polynomial time algorithm for treewidth and minimum fill-in, respectively, when restricted to the class \(\mathcal{G}\). Many interesting classes of intersection graphs have a polynomial time algorithm computing all minimal separators, like permutation graphs, circle graphs, circular arc graphs, distance hereditary graphs, chordal bipartite graphs etc. Our result generalizes earlier results for the treewidth and minimum fill-in for several of these classes. We also consider the related problems pathwidth and interval completion when restricted to some special graph classes.

Additive Tree Spanners

A spanning tree of a graph is a k-additive tree spanner whenever the distance of every two vertices in the graph differs from that in the tree by at most k. In this paper we show that certain classes of graphs, such as distance-hereditary graphs, interval graphs, asteroidal triple-free graphs, allow some constant k such that every member in the class has some k-additive tree spanner. On the other hand, there are chordal graphs without a k-additive tree spanner for arbitrarily large k.