Mathematics and Statistics Faculty Publications

Self-Assessment: Aiding Awareness of Achievement

David E. Brown — Mon, 18 Mar 2013 15:03:22 PDT

The Development and Evaluation of a Program for Improving and Assessing the Teaching of Mathematics and Statistics

David E. Brown et al. — Mon, 18 Mar 2013 15:03:22 PDT

Assessing Proofs with Rubrics: The RVF Method

David E. Brown — Mon, 18 Mar 2013 15:03:21 PDT

Probe Interval Orders

David E. Brown et al. — Mon, 18 Mar 2013 15:03:20 PDT

Variations on Interval Graphs

David E. Brown et al. — Mon, 18 Mar 2013 15:03:19 PDT

Characterizations of Interval Bigraphs and Unit Interval Bigraphs

David E. Brown et al. — Mon, 18 Mar 2013 15:03:18 PDT

Interval k-graphs

David E. Brown et al. — Mon, 18 Mar 2013 15:03:17 PDT

Relationships Among Classes of Interval Bigraphs, (0,1)-matrices, and Circular Arc Graphs

David E. Brown et al. — Mon, 18 Mar 2013 15:03:16 PDT

On Cycle and Bi-cycle Extendability in Chordal and Chordal Bipartite Graphs

LeRoy B. Beasley et al. — Mon, 18 Mar 2013 15:03:14 PDT

Bipartite Probe Interval Graphs, Interval Point Bigraphs, and Circular ArcGraphs

David E. Brown et al. — Mon, 18 Mar 2013 15:03:13 PDT

A Characterization of Cycle-free Unit Probe Interval Graphs

David E. Brown et al. — Mon, 18 Mar 2013 15:03:12 PDT

A graph is a probe interval graph (PIG) if its vertices can be partitioned into probes and nonprobes with an interval assigned to each vertex so that vertices are adjacent if and only if their corresponding intervals overlap and at least one of them is a probe. PIGs are a generalization of interval graphs introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. PIGs have been characterized in the cycle-free case by Sheng, and other miscellaneous results are given by McMorris, Wang, and Zhang. Johnson and Spinrad give a polynomial time recognition algorithm for when the partition of vertices into probes and nonprobes is given. The complexity for the general recognition problem is not known. Here, we restrict attention to the case where all intervals have the same length, that is, we study the unit probe interval graphs and characterize the cycle-free graphs that are unit probe interval graphs via a list of forbidden induced subgraphs.

Interval Tournaments

David E. Brown et al. — Mon, 18 Mar 2013 15:03:12 PDT

A tournament is an orientation of a complete graph. A directed graph is an interval digraph if for each vertex v there corresponds an ordered pair of intervals (Sv, Tv) such that u → v if and only if Su ∩ Tv ≠ ∅. A bipartite graph is an interval bigraph if to each vertex there corresponds an interval such that vertices are adjacent if and only if their corresponding intervals intersect and each vertex belongs to a different partite set. We use the equivalence of the models for interval digraphs and interval bigraphs to characterize tournaments that are interval digraphs via forbidden subtournaments and prove that a tournament on n vertices is an interval digraph if and only if it has a transitive (n − 1)-subtournament. We also characterize the obstructions to the existence of a transitive subtournament of order n − 1 in a tournament of order n.

Extending Partial Tournanents

LeRoy B. Beasley et al. — Mon, 18 Mar 2013 15:03:11 PDT

Let A be a (0,1,∗)-matrix with main diagonal all 0’s and such that if ai,j=1 or ∗ then aj,i=∗ or 0. Under what conditions on the row sums, and or column sums, of A is it possible to change the ∗’s to 0’s or 1’s and obtain a tournament matrix (the adjacency matrix of a tournament) with a specified score sequence? We answer this question in the case of regular and nearly regular tournaments. The result we give is best possible in the sense that no relaxation of any condition will always yield a matrix that can be so extended.

Preserving Regular Tournaments and Term Rank-1

LeRoy B. Beasley et al. — Mon, 18 Mar 2013 15:03:10 PDT

We investigate linear operators which map certain types of tournaments to themselves. To this end we also characterize term rank-1 preservers on the set of matrices whose associated digraphs are simple loopless directed graphs, and find that this set of operators is more diverse than might be expected.

Embedding Tournaments

LeRoy B. Beasley et al. — Mon, 18 Mar 2013 15:03:09 PDT

Cycle Extendability in Graphs and Digraphs

LeRoy B. Beasley et al. — Mon, 18 Mar 2013 15:03:08 PDT

In 1990, Hendry conjectured that all chordal Hamiltonian graphs are cycle extendable, that is, the vertices of each non-Hamiltonian cycle are contained in a cycle of length one greater. Let A be a symmetric (0,1)-matrix with zero main diagonal such that A is the adjacency matrix of a chordal Hamiltonian graph. Hendry’s conjecture in this case is that every k×k principle submatrix of A that dominates a full cycle permutation k×k matrix is a principle submatrix of a (k+1)×(k+1) principle submatrix of A that dominates a (k+1)×(k+1) full cycle permutation matrix. This article generalizes the concept of cycle-extendability to S-extendable; that is, with S⊆{1,2,…,n} and G a graph on n vertices, G is S-extendable if the vertices of every non-Hamiltonian cycle are contained in a cycle length i greater, where i∈S. We investigate this concept in directed graphs and in particular tournaments, i.e., anti-symmetric matrices with zero main diagonal.

Linear Time Recognition Algorithms and Structure Theorems for Bipartite Tolerance and Bipartite Probe Interval Graphs

David E. Brown et al. — Mon, 18 Mar 2013 15:03:07 PDT

A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V,E) is a tolerance graph if each vertex v ∈V can be associated to an interval Iv of the real line and a positive real number tv such that uv ∈E if and only if |Iu ∩Iv| ≥ min (tu,tv). In this paper we present O(|V| + |E|) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms.

Boolean Rank of Upset Tournament Matrices

David E. Brown et al. — Mon, 18 Mar 2013 15:03:06 PDT

The Boolean rank of an m×n(0,1)-matrix M is the minimum k for which matrices A and B exist with M=AB, A is m×k, B is k×n, and Boolean arithmetic is used. The intersection number of a directed graph D is the minimum cardinality of a finite set S for which each vertex v of D can be represented by an ordered pair (Sv,Tv) of subsets of S such that there is an arc from vertex u to vertex v in D if and only if Su∩Tv≠Ø. The intersection number of a digraph is equal to the Boolean rank of its adjacency matrix. Using this fact, we show that the intersection number of an upset tournament, equivalently, the Boolean rank of its adjacency matrix, is equal to the number of maximal subpaths of certain types in its upset path.

Forbidden Subgraph Characterization of Bipartite Unit Probe Interval Graphs

David E. Brown et al. — Mon, 18 Mar 2013 15:03:05 PDT