60 research outputs found

    Decomposition theorem on matchable distributive lattices

    Full text link
    A distributive lattice structure M(G){\mathbf M}(G) has been established on the set of perfect matchings of a plane bipartite graph GG. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph GG is elementary, then M(G){\mathbf M}(G) is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice L\mathbf{L} is an MDL if and only if each factor in any cartesian product decomposition of L\mathbf{L} is an MDL. Two types of MDLs are presented: J(m×n)J(\mathbf{m}\times \mathbf{n}) and J(T)J(\mathbf{T}), where m×n\mathbf{m}\times \mathbf{n} denotes the cartesian product between mm-element chain and nn-element chain, and T\mathbf{T} is a poset implied by any orientation of a tree.Comment: 19 pages, 7 figure

    Resonance graphs of plane bipartite graphs as daisy cubes

    Full text link
    We characterize all plane bipartite graphs whose resonance graphs are daisy cubes and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we prove that if GG is a plane elementary bipartite graph other than K2K_2, then the resonance graph R(G)R(G) is a daisy cube if and only if the Fries number of GG equals the number of finite faces of GG, which in turn is equivalent to GG being homeomorphically peripheral color alternating. Next, we extend the above characterization from plane elementary bipartite graphs to all plane bipartite graphs and show that the resonance graph of a plane bipartite graph GG is a daisy cube if and only if GG is weakly elementary bipartite and every elementary component of GG other than K2K_2 is homeomorphically peripheral color alternating. Along the way, we prove that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes

    Plane augmentation of plane graphs to meet parity constraints

    Get PDF
    A plane topological graph G=(V, E) is a graph drawn in the plane whose vertices are points in the plane and whose edges are simple curves that do not intersect, except at their endpoints. Given a plane topological graph G=(V, E) and a set CG of parity constraints, in which every vertex has assigned a parity constraint on its degree, either even or odd, we say that G is topologically augmentable to meet CG if there exists a set E' of new edges, disjoint with E, such that G'=(V, E¿E') is noncrossing and meets all parity constraints. In this paper, we prove that the problem of deciding if a plane topological graph is topologically augmentable to meet parity constraints is NP-complete, even if the set of vertices that must change their parities is V or the set of vertices with odd degree. In particular, deciding if a plane topological graph can be augmented to a Eulerian plane topological graph is NP-complete. Analogous complexity results are obtained, when the augmentation must be done by a plane topological perfect matching between the vertices not meeting their parities. We extend these hardness results to planar graphs, when the augmented graph must be planar, and to plane geometric graphs (plane topological graphs whose edges are straight-line segments). In addition, when it is required that the augmentation is made by a plane geometric perfect matching between the vertices not meeting their parities, we also prove that this augmentation problem is NP-complete for plane geometric paths. For the particular family of maximal outerplane graphs, we characterize maximal outerplane graphs that are topological augmentable to satisfy a set of parity constraints. We also provide a polynomial time algorithm that decides if a maximal outerplane graph is topologically augmentable to meet parity constraints, and if so, produces a set of edges with minimum cardinality

    Dominating Sets in Plane Triangulations

    Get PDF
    In 1996, Matheson and Tarjan conjectured that any n-vertex triangulation with n sufficiently large has a dominating set of size at most n/4. We prove this for graphs of maximum degree 6.Comment: 14 pages, 6 figures; Revised lemmas 6-8, clarified arguments and fixed typos, result unchange
    corecore