4 research outputs found
Uniformly dissociated graphs
Creative Commons Attribution 3.0 International LicenseA set D of vertices in a graph G is called a dissociation set if every vertex in D has at most one neighbor in D. We call a graph G uniformly dissociated if all maximal dissociation sets are of the same cardinality. Characterizations of uniformly dissociated graphs with small cardinalities of dissociation sets are proven; in particular, the graphs in which all maximal dissociation sets are of cardinality 2 are the complete graphs on at least two vertices from which possibly a matching is removed, while the graphs in which all maximal dissociation sets are of cardinality 3 are the complements of the K4-free geodetic graphs with diameter 2. A general construction by which any graph can be embedded as an induced sub graph of a uniformly dissociated graph is also presented. In the main result we characterize uniformly dissociated graphs with girth at least 7 to be either isomorphic to C7, or obtainable from an arbitrary graph H with girth at least 7 by identifying each vertex of H with a leaf of a copy of P3
General -position sets
The general -position number of a graph is the
cardinality of a largest set for which no three distinct vertices from
lie on a common geodesic of length at most . This new graph parameter
generalizes the well studied general position number. We first give some
results concerning the monotonic behavior of with respect to
the suitable values of . We show that the decision problem concerning
finding is NP-complete for any value of . The value of when is a path or a cycle is computed and a structural
characterization of general -position sets is shown. Moreover, we present
some relationships with other topics including strong resolving graphs and
dissociation sets. We finish our exposition by proving that is
infinite whenever is an infinite graph and is a finite integer.Comment: 16 page
2018 Furman University Faculty Scholarship Reception Program
On February 23, 2018, the Libraries and the Office of the Provost hosted Furman’s second Faculty Scholarship Reception to recognize and celebrate the scholarly publications and creative works of Furman faculty members. The reception, held in the Blackwell Atrium of the James B. Duke Library, showcased scholarship published by 70 faculty members during the 2017 calendar year. The following faculty provided four-minute presentations on their scholarly or creative works: Omar Camenates, Associate Professor, Music Kevin Hutson, Professor, and Liz Bouzarth, Associate Professor, Mathematics Amy Jonason, Assistant Professor, Sociology Bill Pierce, Professor, and Scott Murr, Assistant Professor, Health Sciences Greg Springsteen, Associate Professor, Chemistr