arXiv subjects

Michael D. Barrus

Publications and source records attributed to Michael D. Barrus.

21 records · Page 2Linked to original sources

arXiv2013

Minimal forbidden sets for degree sequence characterizations

Given a set $\mathcal{F}$ of graphs, a graph $G$ is $\mathcal{F}$-free if $G$ does not contain any member of $\mathcal{F}$ as an induced subgraph. Barrus, Kumbhat, and Hartke [M. D. Barrus, M. Kumbhat, and S. G. Hartke, Graph classes characterized both by forbidden subgraphs and degree sequences, J. Graph Theory (2008), no. 2, 131--148] called $\mathcal{F}$ a degree-sequence-forcing (DSF) set if, for each graph $G$ in the class $\mathcal{C}$ of $\mathcal{F}$-free graphs, every realization of the degree sequence of $G$ is also in $\mathcal{C}$. A DSF set is minimal if no proper subset is also DSF. In this paper, we present new properties of minimal DSF sets, including that every graph is in a minimal DSF set and that there are only finitely many DSF sets of cardinality $k$. Using these properties and a computer search, we characterize the minimal DSF triples.

math.CO↗

arXiv2013

Hereditary unigraphs and Erdős--Gallai equalities

We give characterizations of the structure and degree sequences of hereditary unigraphs, those graphs for which every induced subgraph is the unique realization of its degree sequence. The class of hereditary unigraphs properly contains the threshold and matrogenic graphs, and the characterizations presented here naturally generalize those known for these other classes of graphs. The degree sequence characterization of hereditary unigraphs makes use of the list of values $k$ for which the $k$th Erdős--Gallai inequality holds with equality for a graphic sequence. Using the canonical decomposition of Tyshkevich, we show how this list describes structure common among all realizations of an arbitrary graphic sequence.

math.CO↗

arXiv2011

On 2-switches and isomorphism classes

A 2-switch is an edge addition/deletion operation that changes adjacencies in the graph while preserving the degree of each vertex. A well known result states that graphs with the same degree sequence may be changed into each other via sequences of 2-switches. We show that if a 2-switch changes the isomorphism class of a graph, then it must take place in one of four configurations. We also present a sufficient condition for a 2-switch to change the isomorphism class of a graph. As consequences, we give a new characterization of matrogenic graphs and determine the largest hereditary graph family whose members are all the unique realizations (up to isomorphism) of their respective degree sequences.

math.CO↗