There is only one finite, 2-connected, linearly convex graph in the Archimedean triangular tiling that does not have a Hamiltonian cycle. The vertices and polygonal edges of the planar Archimedean tilings 44 and 36 of the plane, partially shown in Figs. 1 and 2, respectively, are called the square tiling graph (STG) and the triangular tiling graph (TTG). (See [1].) A subgraph G of TTG is linear...

We call a graph matching-covered if every line belongs to a perfect matching. We study the technique of "ear-decompositions" of such graphs. We prove that a non-bipartite matchingcovered graph contains K~ or K2@Ka (the triangular prism). Using this result, we give new characterizations of those graphs whose matching and covering numbers are equal. We apply these results to the theory of r-criti...

We introduce a new algorithm to convert triangular meshes of polygonal regions, with or without holes, into strictly convex quadrilateral meshes of small bounded size. Our algorithm includes all vertices of the triangular mesh in the quadrilateral mesh, but may add extra vertices (called Steiner points). We show that if the input triangular mesh has t triangles, our algorithm produces a mesh wi...

Given two triangular embeddings f and f ′ of a complete graph K and given a bijection φ : V (K) → V (K), denote by M(φ) the set of faces (x, y, z) of f such that (φ(x), φ(y), φ(z)) is not a face of f . The distance between f and f ′ is the minimal value of |M(φ)| as φ ranges over all bijections between the vertices of K. We consider the minimal nonzero distance between two triangular embeddings...

A supergrid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional supergrid. The supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. The Hamiltonian cycle problem for grid and triangular grid graphs was known to be NP-complete. In the past, we have shown that the Hamiltonian cycle problem for supergrid graphs is also NP-complete. ...

The main purpose of the paper is to develop an approach to evaluation or estimation of the spanning tree congestion of planar graphs. This approach is used to evaluate the spanning tree congestion of triangular grids.

This paper describes the automatic generation of a structureddominant mesh based on an automatic block decomposition obtained using the medial object. The mesh generator will produce a mesh for an arbitrary two-dimensional geometry, where quadrilateral-dominant or triangular meshing is used when structured meshing fails. The domain is automatically partitioned into blocks (sub-regions), and an ...

The decomposition of a linkage into minimal components is a central tool of analysis and synthesis of linkages. In this paper we prove that every pinned d-isostatic (minimally rigid) graph (grounded linkage) has a unique decomposition into minimal strongly connected components (in the sense of directed graphs), or equivalently intominimal pinned isostatic graphs, which we call d-Assur graphs. W...