Ruth Haas

Planar minimally rigid graphs and pseudo-triangulations

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than p). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide---to the best of our knowledge---the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.

The k-Dominating Graph

Given a graph G, the k-dominating graph of G, Dk(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in Dk(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph Dk(G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of Dk(G). In this paper we give conditions that ensure Dk(G) is connected.

Partial list colorings

Abstract Suppose G is an s -choosable graph with n vertices, and every vertex of G is assigned a list of t colors. We conjecture that at least (t/s)n of the vertices of G can be colored from these lists. We provide lower bounds and consider related questions. For instance, we show that if G is χ -colorable (rather than being s -choosable), then more than (1−((χ−1)/χ) t )n of the vertices of G can be colored from the lists and that this is asymptotically best possible. We include a number of open questions.

Parsimonious edge coloring

Abstract In a graph G of maximum degree Δ, let γ denote the largest fraction of edges that can be Δ-edge-colored. This paper investigates lower bounds for γ together with infinite families of graphs in which γ is bounded away from 1. For instance, if G is cubic, then γ ⩾ 13 15 ; and there exists an infinite family of 3-connected cubic graphs in which γ ⩽ 25 27 .

Bounds on the signed domination number of a graph

Abstract Let G = (V, E) be a simple graph on vertex set V and define a function f: V → {−1, 1}. The function f is a signed dominating function if for every vertex x ∈ V, the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed domination number of G γs (G) is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed domination number of a general graph with a given minimum and maximum degree, generalizing a number of previously known results. Using similar techniques we give upper and lower bounds for the signed domination number of some simple graph products: the grid P j × P k , C j × P k and C j × C k . For fixed width, these bounds differ by only a constant.

Signed domination numbers of a graph and its complement

Abstract Let G=(V,E) be a simple graph on vertex set V and define a function f:V→{−1,1}. The function f is a signed dominating function if for every vertex x∈V, the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. Let G denote the complement of G. In this paper we establish upper and lower bounds on γ s (G)+γ s ( G ) .

Star forests, dominating sets and Ramsey-type problems

A star forest of a graph G is a spanning subgraph of G in which each component is a star. The minimum number of edges required to guarantee that an arbitrary graph, or a bipartite graph, has a star forest of size n is determined. Sharp lower bounds on the size of a largest star forest are also determined. For bipartite graphs, these are used to obtain an upper bound on the domination number in terms of the number of vertices and edges in the graph, which is an improvement on a bound of Vizing. In turn, the results on bipartite graphs are used to determine the minimum number of lattice points required so that there exists a subset of n lattice points, no three of which form a right triangle with legs parallel to the coordinate axes.

The canonical coloring graph of trees and cycles

For a graph G and an ordering of the vertices π , the set of canonical k -colorings of G under π is the set of non-isomorphic proper k -colorings of G that are lexicographically least under π . The canonical coloring graph Can π k ( G ) is the graph with vertex set the canonical colorings of G and two vertices are adjacent if the colorings differ in exactly one place. This is a natural variation of the color graph C k ( G ) where all colorings are considered. We show that every graph has a canonical coloring graph which is disconnected; that trees have canonical coloring graphs that are Hamiltonian; and cycles have canonical coloring graphs that are connected.

Counting edge-Kempe-equivalence classes for 3-edge-colored cubic graphs

Abstract Two n -edge colorings of a graph are edge-Kempe equivalent if one can be obtained from the other by a series of edge-Kempe switches. In this work we show every planar bipartite cubic graph has exactly one edge-Kempe equivalence class, when 3 = χ ′ ( G ) colors are used. In contrast, we also exhibit infinite families of nonplanar bipartite cubic (and thus 3-edge colorable) graphs with a range of numbers of edge-Kempe equivalence classes when using 3 colors. These results address a question raised by Mohar.

Module and vector space bases for spline spaces

Abstract For Δ, a triangulated d-dimensional region in Rd, let Smr(Δ) be the vector space of all Cr functions F on Δ such that for any simplex σ ϵ Δ, F¦ σ is a polynomial of degree at most m. Smr(Δ) is the often studied vector space of splines on Δ of degree m and smoothness r. We define Sr(Δ) = ∨m Smr(Δ). Sr(Δ) is a module over the polynomial ring R [x1,…, xd]. In certain cases a module basis for Sr(Δ) provides vector space bases for the corresponding Smr(Δ) via simple linear algebra. In this work we examine that relationship and consider techniques for finding module bases of spaces Sr(Δ). A basis for Sr(Δ) is reduced if every element F in Sr(Δ) can be represented using only basis elements of degree less than the degree of F. We show the relationship between the dimension of the spaces Smr(Δ) and the degrees of the reduced basis elements of Sr(Δ). Ths result leads to techniques for finding module bases. These techniques are used to find module bases for spline spaces on cross-cut grids.