On maximal chain subgraphs and covers of bipartite graphs

In this paper, we address three related problems. One is the enumeration of all the maximal edge induced chain subgraphs of a bipartite graph, for which we provide a polynomial delay algorithm. We give bounds on the number of maximal chain subgraphs for a bipartite graph and use them to establish the input-sensitive complexity of the enumeration problem. The second problem we treat is the one of finding the minimum number of chain subgraphs needed to cover all the edges a bipartite graph. For this we provide an exact exponential algorithm with a non trivial complexity. Finally, we approach the problem of enumerating all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time

On Maximal Chain Subgraphs and Covers of Bipartite Graphs

International audienceIn this paper, we address three related problems. One is the enumeration of all the maximal edge induced chain subgraphs of a bipartite graph, for which we provide a polynomial delay algorithm. We give bounds on the number of maximal chain subgraphs for a bipartite graph and use them to establish the input-sensitive complexity of the enumeration problem. The second problem we treat is the one of finding the minimum number of chain subgraphs needed to cover all the edges a bipartite graph. For this we provide an exact exponential algorithm with a non trivial complexity. Finally, we approach the problem of enumerating all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time

Matchings, coverings, and Castelnuovo-Mumford regularity

We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

Algorithms for the quantitative Lock/Key model of cytoplasmic incompatibility

Cytoplasmic incompatibility (CI) relates to the manipulation by the parasite Wolbachia of its host reproduction. Despite its widespread occurrence, the molecular basis of CI remains unclear and theoretical models have been proposed to understand the phenomenon. We consider in this paper the quantitative Lock-Key model which currently represents a good hypothesis that is consistent with the data available. CI is in this case modelled as the problem of covering the edges of a bipartite graph with the minimum number of chain subgraphs. This problem is already known to be NP-hard, and we provide an exponential algorithm with a non trivial complexity. It is frequent that depending on the dataset, there may be many optimal solutions which can be biologically quite different among them. To rely on a single optimal solution may therefore be problematic. To this purpose, we address the problem of enumerating (listing) all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time. Interestingly, in order to solve the above problems, we considered also the problem of enumerating all the maximal chain subgraphs of a bipartite graph and improved on the current results in the literature for the latter. Finally, to demonstrate the usefulness of our methods we show an application on a real dataset

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure