A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2-Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NP-complete, as is the 1Layer Planarization problem in which the permutation of t...

We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-Pfaffian graphs minimal with respect to the matching minor relation. This is in sharp contrast with the bipartite case, as Little [7] proved that every bipartite non-Pfaffian graph contains a matching minor isomorphic to K3,3. We relax the notion of a matching minor and conjecture that there are onl...

The excess of a graph G is defined as the minimum number of edges that must be deleted from G in order to get a forest. We prove that every graph with excess at most k has chromatic number at most 1 2(3+ √ 1 + 8k) and that this bound is tight. Moreover, we prove that the oriented chromatic number of any graph with excess k is at most k+3, except for graphs having excess 1 and containing a direc...

For-profit institutions have seen considerable growth in the last decade, and much of that growth has been accomplished with an expansion of online course offerings. Faculty satisfaction is a critical element in any academic institution, and this study looked at the influence of five demographic variables (faculty status, age, teaching experience, gender, and language status) on satisfaction fo...

The structure of previous known infinite families of crossing–critical graphs had led to the conjecture that crossing–critical graphs have bounded bandwidth. If true, this would imply that crossing–critical graphs have bounded degree, that is, that they cannot contain subdivisions of K1,n for arbitrarily large n. In this paper we prove two new results that revolve around this question. On the p...

A critical element of most national telecom policy objectives is advancing universal service. International Telecommunications Union (ITU) considers universal service provision as purely for the purpose of making available telecommunication services of specified quality, and in light of the specific national conditions, at an affordable price to that minority of potential users who do not have ...

A graph G is inexhaustible if whenever a vertex of G is deleted the remaining graph is isomorphic to G. We address a question of Cameron [6], who asked which countable graphs are inexhaustible. In particular, we prove that there are continuum many countable inexhaustible graphs with properties in common with the infinite random graph, including adjacency properties and universality. Locally fin...

A connected graph G is said to be factor-critical if G − v has a perfect matching for every vertex v of G. Lovász proved that every factor-critical graph has an ear decomposition. In this paper, the ear decomposition of the factor-critical graphs G satisfying that G− v has a unique perfect matching for any vertex v of G with degree at least 3 is characterized. From this, the number of maximum m...

We introduce a graph-theoretic dissolution model that applies to a number of redistribution scenarios such as gerrymandering in political districting or work balancing in an online situation. The central aspect of our model is the deletion of certain vertices and the redistribution of their loads to neighboring vertices in a perfectly balanced way. We investigate how the underlying graph struct...