60 research outputs found
Decomposition theorem on matchable distributive lattices
A distributive lattice structure has been established on the
set of perfect matchings of a plane bipartite graph . 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 is elementary, then is
irreducible. Based on this result, a decomposition theorem on MDLs is obtained:
a finite distributive lattice is an MDL if and only if each factor
in any cartesian product decomposition of is an MDL. Two types of
MDLs are presented: and , where
denotes the cartesian product between -element
chain and -element chain, and is a poset implied by any
orientation of a tree.Comment: 19 pages, 7 figure
Resonance graphs of plane bipartite graphs as daisy cubes
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 is a plane elementary bipartite
graph other than , then the resonance graph is a daisy cube if and
only if the Fries number of equals the number of finite faces of , which
in turn is equivalent to 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 is a daisy cube if and only if is
weakly elementary bipartite and every elementary component of other than
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
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
Recommended from our members
Spatial arrangements in architecture and mechanical engineering: some aspects of their representation and construction
Spatial arrangements in architecture and mechanical engineering are represented by incidence structures and classified according to properties of these incidence structures. The relationships between classes are given by ornamentation operations and the construction of elements in fundamental classes by substructure replacement operations. Thus representations of the spatial arrangements for possible designs are generated.
Planar maps represent spatial arrangements in architecutral plans. The edges correspond to walls and vertices to incidence between walls. Plans represented by 3-vertex connected maps are ornamented by rooting and extension operations. Further ornamentation specifies access between regions. Plans with all regions adjacent to the exterior correspond to outerplane maps. Trivalent maps represent an important class of plans. Fundamental plans with r internal regions and s regions adjacent to the exterior are represented by [r,s] triangulations. Ornamentations of simple [r,s] triangulations are specified which represent plans with rectangular regions. Plans with walls aligned along two directions are represented by rectangular shapes whose maximal lines correspond to contiguous aligned walls. Rules of construction for various classes are given and the incidence structures of maximal lines and regions are characterized.
Spatial arrangements in machines are represented by systems whose blocks correspond to links and vertices to joints. The dual systems are also used. Coplanar kinematic chains with revolute pairs are classified according to mobility and connectedness. Two fundamental classes are considered. First, the chains with binary joints, represented by simple graphs and constructed by two new methods: (i) suspended chain and cycle addition and (ii) subgraph replacement. Second, the chains with binary links which are constructed by subgraph replacement
Dominating Sets in Plane Triangulations
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
- …