Completing Partial Packings of Bipartite Graphs

Given a bipartite graph $H$ and an integer $n$, let $f(n;H)$ be the smallest integer such that, any set of edge disjoint copies of $H$ on $n$ vertices, can be extended to an $H$-design on at most $n+f(n;H)$ vertices. We establish tight bounds for the growth of $f(n;H)$ as $n \rightarrow \infty$. In particular, we prove the conjecture of F\"uredi and Lehel \cite{FuLe} that $f(n;H) = o(n)$. This settles a long-standing open problem

Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions

We attach to each finite bipartite separated graph (E,C) a partial dynamical system (\Omega(E,C), F, \theta), where \Omega(E,C) is a zero-dimensional metrizable compact space, F is a finitely generated free group, and {\theta} is a continuous partial action of F on \Omega(E,C). The full crossed product C*-algebra O(E,C) = C(\Omega(E,C)) \rtimes_{\theta} F is shown to be a canonical quotient of the graph C*-algebra C^*(E,C) of the separated graph (E,C). Similarly, we prove that, for any *-field K, the algebraic crossed product L^{ab}_K(E,C) = C_K(\Omega(E,C)) \rtimes_\theta^{alg} F is a canonical quotient of the Leavitt path algebra L_K(E,C) of (E,C). The monoid V(L^{ab}_K(E,C)) of isomorphism classes of finitely generated projective modules over L^{ab}_K(E,C) is explicitly computed in terms of monoids associated to a canonical sequence of separated graphs. Using this, we are able to construct an action of a finitely generated free group F on a zero-dimensional metrizable compact space Z such that the type semigroup S(Z, F, K) is not almost unperforated, where K denotes the algebra of clopen subsets of Z. Finally we obtain a characterization of the separated graphs (E,C) such that the canonical partial action of F on \Omega(E,C) is topologically free.Comment: Final version to appear in Advances in Mathematic

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

Covering Partial Cubes with Zones

A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite poset. For several such special cases, we give upper and lower bounds on the minimum size of a covering by zones. We also consider the computational complexity of those problems, and establish some hardness results

Interdiction Problems on Planar Graphs

Interdiction problems are leader-follower games in which the leader is allowed to delete a certain number of edges from the graph in order to maximally impede the follower, who is trying to solve an optimization problem on the impeded graph. We introduce approximation algorithms and strong NP-completeness results for interdiction problems on planar graphs. We give a multiplicative $(1 + \epsilon)$-approximation for the maximum matching interdiction problem on weighted planar graphs. The algorithm runs in pseudo-polynomial time for each fixed $\epsilon > 0$. We also show that weighted maximum matching interdiction, budget-constrained flow improvement, directed shortest path interdiction, and minimum perfect matching interdiction are strongly NP-complete on planar graphs. To our knowledge, our budget-constrained flow improvement result is the first planar NP-completeness proof that uses a one-vertex crossing gadget.Comment: 25 pages, 9 figures. Extended abstract in APPROX-RANDOM 201