Charis Papadopoulos

Enumerating Minimal Subset Feedback Vertex Sets

The Subset Feedback Vertex Set problem takes as input a pair (G,S), where G=(V,E) is a graph with weights on its vertices, and S⊆V. The task is to find a set of vertices of total minimum weight to be removed from G, such that in the remaining graph no cycle contains a vertex of S. We show that this problem can be solved in time O(1.8638n), where n=|V|. This is a consequence of the main result of this paper, namely that all minimal subset feedback vertex sets of a graph can be enumerated in time O(1.8638n).

Characterising the linear clique-width of a class of graphs by forbidden induced subgraphs ☆

Abstract We study the linear clique-width of graphs that are obtained from paths by disjoint union and adding true twins. We show that these graphs have linear clique-width at most 4, and we give a complete characterisation of their linear clique-width by forbidden induced subgraphs. As a consequence, we obtain a linear-time algorithm for computing the linear clique-width of the considered graphs. Our results extend the previously known set of forbidden induced subgraphs for graphs of linear clique-width at most 3.

Graphs of linear clique-width at most 3

Abstract A graph has linear clique-width at most k if it has a clique-width expression using at most k labels such that every disjoint union operation has an operand which is a single vertex graph. We give the first characterisation of graphs of linear clique-width at most 3, and we give the first polynomial-time recognition algorithm for graphs of linear clique-width at most 3. In addition, we present new characterisations of graphs of linear clique-width at most 2. We also give a layout characterisation of graphs of bounded linear clique-width; a similar characterisation was independently shown by Gurski and by Lozin and Rautenbach.

Clustering with partial information

The Correlation Clustering problem, also known as the Cluster Editing problem, seeks to edit a given graph by adding and deleting edges to obtain a collection of vertex-disjoint cliques, such that the editing cost is minimized. The Edge Clique Partitioning problem seeks to partition the edges of a given graph into edge-disjoint cliques, such that the number of cliques is minimized. Both problems are known to be NP-hard, and they have been previously studied with respect to approximation and fixed-parameter tractability. In this paper we study these two problems in a more general setting that we term fuzzy graphs, where the input graphs may have missing information, meaning that whether or not there is an edge between some pairs of vertices of the input graph can be undecided. For fuzzy graphs the Correlation Clustering and Edge Clique Partitioning problems have previously been studied only with respect to approximation. Here we give parameterized algorithms based on kernelization for both problems. We prove that the Correlation Clustering problem is fixed-parameter tractable on fuzzy graphs when parameterized by (k,r), where k is the editing cost and r is the minimum number of vertices required to cover the undecided edges. In particular we show that it has a polynomial-time reduction to a problem kernel on O(k^2+r) vertices. We provide an analogous result for the Edge Clique Partitioning problem on fuzzy graphs. Using (k,r) as parameters, where k bounds the size of the partition, and r is the minimum number of vertices required to cover the undecided edges, we describe a polynomial-time kernelization to a problem kernel on O(k^4@?3^r) vertices. This implies fixed-parameter tractability for this parameterization. Furthermore we also show that parameterizing only by the number of cliques k, is not enough to obtain fixed-parameter tractability. The problem remains, in fact, NP-hard for each fixed k>2.

The Number of Spanning Trees in K n -Complements of Quasi-Threshold Graphs

Abstract.In this paper we examine the classes of graphs whose Kn-complements are trees or quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of Kn, the Kn-complement of H is the graph Kn−H which is obtained from Kn by removing the edges of H. Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of the graph. Our results generalize previous results and extend the family of graphs of the form Kn−H admitting formulas for the number of their spanning trees.

Single-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletions

We study graph properties that admit an increasing, or equivalently decreasing, sequence of graphs on the same vertex set such that for any two consecutive graphs in the sequence their difference is a single edge. This is useful for characterizing and computing minimal completions and deletions of arbitrary graphs into having these properties. We prove that threshold graphs and chain graphs admit such sequences. Based on this characterization and other structural properties, we present linear-time algorithms both for computing minimal completions and deletions into threshold, chain, and bipartite graphs, and for extracting a minimal completion or deletion from a given completion or deletion. Minimum completions and deletions into these classes are NP-hard to compute.

A new representation of proper interval graphs with an application to clique-width☆

Abstract We introduce a new representation of proper interval graphs that can be computed in linear time and stored in O ( n ) space. This representation is a 2-dimensional vertex partition. It is particularly interesting with respect to clique-width. Based on this representation, we prove new upper bounds on the clique-width of proper interval graphs.

Minimal comparability completions of arbitrary graphs

A transitive orientation of an undirected graph is an assignment of directions to its edges so that these directed edges represent a transitive relation between the vertices of the graph. Not every graph has a transitive orientation, but every graph can be turned into a graph that has a transitive orientation, by adding edges. We study the problem of adding an inclusion minimal set of edges to an arbitrary graph so that the resulting graph is transitively orientable. We show that this problem can be solved in polynomial time, and we give a surprisingly simple algorithm for it. We use a vertex incremental approach in this algorithm, and we also give a more general result that describes graph classes @P for which @P completion of arbitrary graphs can be achieved through such a vertex incremental approach.

On the performance of the First-Fit coloring algorithm on permutation graphs

Abstract In this paper we study the performance of a particular on-line coloring algorithm, the First-Fit or Greedy algorithm, on a class of perfect graphs namely the permutation graphs. We prove that the largest number of colors χ FF (G) that the First-Fit coloring algorithm (FF) needs on permutation graphs of chromatic number χ(G)=χ when taken over all possible vertex orderings is not linearly bounded in terms of the off-line optimum, if χ is a fixed positive integer. Specifically, we prove that for any integers χ>0 and k≥0 , there exists a permutation graph G on n vertices such that χ(G)=χ and χ FF (G)≥ 1 2 ((χ 2 +χ)+k(χ 2 −χ)) , for sufficiently large n . Our result shows that the class of permutation graphs P is not First-Fit χ -bounded; that is, there exists no function f such that for all graphs G∈ P , χ FF (G)≤f(ω(G)) . Recall that for perfect graphs ω(G)=χ(G) , where ω(G) denotes the clique number of G .

A Complete Characterisation of the Linear Clique-Width of Path Powers

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.