DISCOVERING PAIRWISE COMPATIBILITY GRAPHS

On graphs that are not PCGs

A graph G = (V, E) is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> such that each leaf l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> of T corresponds to a vertex u ∈ V and there is an edge (u, v) ∈ E if and only if d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> ≤ d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> (l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> , l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> ) ≤ d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> where d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> (l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> , l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> ) is the sum of weights of the edges on the unique path from l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> to l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> in T. In this note, firstly, we show a class of bipartite graphs not to be PCG, that is, it is not possible to draw a pairwise compatibility tree for these graphs. Further we construct a class of more general non-PCGs each member of which contains a bipartite graph belonging to the class of graphs mentioned above as a subgraph. Secondly, we show by computational means that all the bipartite graphs with at most eight vertices are PCGs. In particular all these graphs are PCGs of a particular structure of tree called centipede.

A graph \(G=(V,E)\) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers \(d_{min}\) and \(d_{max}\), \(d_{min} \le d_{max}\), such that each node \(u \in V\) is uniquely associated to a leaf of T and there is an edge \((u,v) \in E\) if and only if \(d_{min} \le d_{T} (u, v) \le d_{max}\), where \(d_{T} (u, v)\) is the sum of the weights of the edges on the unique path \(P_{T}(u,v)\) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. \(C^2_8\)), so improving the previously known result concerning the smallest planar graph known not to be PCG.

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

For any class C of bipartite graphs, we define quasi-C to be the class of all graphs G such that every bipartition of G belongs to C. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.

Bipartite bithreshold graphs

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.

Networks derived from the hypercube are noted for their efficiency in simulating any bounded-degree communication network and hence they play a significant role in parallel machines. For a graph [Formula: see text], a geodetic cover is a subset [Formula: see text] of [Formula: see text], satisfying the condition that every vertex of [Formula: see text] other than the vertices in [Formula: see text] are covered using geodesics connecting any two vertices in [Formula: see text]. The geodetic number of [Formula: see text] is the least order of such covers in [Formula: see text]. This seminal distance-based parameter is well employed in the areas of location theory, convexity theory and game theory. The strong variant and its edge version are recent variations of the problem that were introduced in due of their applications in social networks. The most established and strong hypercube derivative networks are studied in this paper with respect to certain geodetic parameters. Notably, among the networks that we have considered, the butterfly and the Benes networks belong to the class of bipartite graphs for which the geodetic number problem and its strong version are NP-complete. We have determined the geodetic parameters that include the geodetic number, the strong geodetic number and their corresponding edge variants of the networks taken into consideration. In the process, we have identified a new class of graphs, as semi-extreme edge geodesic graphs and that the enhanced butterfly networks belong to this class. The study of such geodetic parameters in the structural models of parallel computation could be useful for the design of efficient algorithms for the structures considered.

Let G and H be any graphs without isolated vertices. The Ramsey set \({\mathcal{R}(G,H)}\) consists of all graphs F without isolated vertices such that \({F \rightarrow (G,H)}\) and \({F-e \nrightarrow (G,H)}\) for every \({e \in E(F)}\). In this paper, we give necessary and sufficient conditions of graphs belonging to the set \({\mathcal{R}(3K_2,K_{1,n})}\), for any n ≥ 3. Furthermore, by using computational approach, we determine all Ramsey \({(3K_2,K_{1,n})}\)-minimal graphs of order at most 10 vertices for \({3 \leq n \leq 7}\). We construct two classes of bipartite graphs in \({\mathcal{R}(3K_2,K_{1,n})}\), for any \({n \geq 3}\). We also present a class of graphs containing a clique of six vertices in this set. We give large classes of graphs in the set \({\mathcal{R}(t K_2, K_{1,n})}\), for \({n \geq 3}\) and \({t \geq 4}\), that are constructed from t disjoint stars.

Characterization and recognition of some opposition and coalition graph classes

Cops and Robbers on intersection graphs

We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. It shows that the best competitive ratio of both deterministic and randomized online algorithms is Theta(log n), where n is the number of vertices of a graph. Furthermore, we prove that this guarantee cannot be improved if an online algorithm has a lookahead of size O(n/log n) or access to a reordering buffer of size n^(1-epsilon), for any 0 < epsilon <= 1. A consequence of our results is that, for all of the above mentioned graph classes except bipartite graphs, the natural First Fit coloring algorithm achieves an optimal performance, up to constant factors, among deterministic and randomized online algorithms.

A proper k-coloring of a graph is an assignment of colors from the set $$\{1,2,\ldots ,k\}$$ to the vertices of the graph such that no two adjacent vertices receive the same color. Given a graph $$G=(V,E)$$ , the Dynamic Coloring problem asks to find a proper k-coloring of G such that for every vertex $$v\in V(G)$$ of degree at least two, there exists at least two distinct colors appearing in the neighborhood of v. The minimum integer k such that there is a dynamic coloring of G using k colors is called the dynamic chromatic number of G and is denoted by $$\chi _d(G)$$ . The problem is NP-complete in general, but solvable in polynomial time on several restricted families of graphs. In this paper, we study the problem on restricted classes of graphs. We show that the problem can be solved in polynomial time on chordal graphs and biconvex bipartite graphs. On the other hand, we show that it is NP-complete on star-convex bipartite graphs, comb-convex bipartite graphs and perfect elimination bipartite graphs. Next, we initiate the study on Dynamic Coloring from the parameterized complexity perspective. First, we show that the problem is fixed-parameter tractable when parameterized by neighborhood diversity or twin-cover. Then, we show that the problem is fixed-parameter tractable when parameterized by the combined parameters clique-width and the number of colors.

Consider the following very simple counting problems associated with a graph G. (i) What is the number of connected subgraphs of G? (ii) How many subgraphs of G are forests? (iii) How many acyclic orientations has G? Each of these is a special case of the general problem of evaluating the Tutte polynomial of a graph (or matroid) at a particular point of the (x, y)-plane — in other words is a Tutte-Gröthendieck invariant. Other invariants include: (iv) the chromatic and flow polynomials of a graph; (v) the partition function of a Q-state Potts model; (vi) the Jones polynomial of an alternating link; (vii) the weight enumerator of a linear code over GF(q). It has been shown that apart from a few special points and 2 special hyperbolae, the exact evaluation of any such invariant is #P-hard even for the very restricted class of planar bipartite graphs. However the question of which points have a fully polynomial randomised approximation scheme is wide open. I shall discuss this problem and give a survey of what is currently known.KeywordsPartition FunctionPlanar GraphLinear CodeConnected SubgraphDense GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures, including as special cases Choquet and Sugeno integrals. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the greatest universal integrals are given. Finally, another construction method for obtaining universal integrals is introduced.