Michael D. Barrus

Degree-associated reconstruction number of graphs

A card of a graph G is a subgraph formed by deleting one vertex. The Reconstruction Conjecture states that each graph with at least three vertices is determined by its multiset of cards. A dacard specifies the degree of the deleted vertex along with the card. The degree-associated reconstruction number drn(G) is the minimum number of dacards that determine G. We show that drn(G)=2 for almost all graphs and determine when drn(G)=1. For k-regular n-vertex graphs, drn(G)@?min{k+2,n-k+1}. For vertex-transitive graphs (not complete or edgeless), we show that drn(G)>=3, give a sufficient condition for equality, and construct examples with large drn. Our most difficult result is that drn(G)=2 for all caterpillars except stars and one 6-vertex example. We conjecture that drn(G)@?2 for all but finitely many trees.

The A 4 -structure of a graph

We define the A4-structure of a graph G to be the 4-uniform hypergraph on the vertex set of G whose edges are the vertex subsets inducing 2K2, C4, or P4. We show that perfection of a graph is determined by its A4-structure. We relate the A4-structure to the canonical decomposition of a graph as defined by Tyshkevich [Discrete Math 220 (2000) 201–238]; for example, a graph is indecomposable if and only if its A4-structure is connected. We also characterize the graphs having the same A4-structure as a split graph.

Neighborhood degree lists of graphs

Abstract The neighborhood degree list (NDL) is a graph invariant that refines information given by the degree sequence and joint degree matrix of a graph and is useful in distinguishing graphs having the same degree sequence. We show that the space of realizations of an NDL is connected via a switching operation. We then determine the NDLs that have a unique realization by a labeled graph; the characterization ties these NDLs and their realizations to the threshold graphs and difference graphs.

Hereditary unigraphs and Erdős–Gallai equalities

Abstract We give characterizations of the structure and degree sequence 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.

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 congurations. We also present a sucient 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.

Colored Saturation Parameters for Rainbow Subgraphs

Inspired by a 1987 result of Hanson and Toft [Edge-colored saturated graphs, J Graph Theory 11 (1987), 191–196] and several recent results, we consider the following saturation problem for edge-colored graphs. An edge-coloring of a graph F is rainbow if every edge of F receives a different color. Let R(F) denote the set of rainbow-colored copies of F. A t-edge-colored graph G is (R(F),t)-saturated if G does not contain a rainbow copy of F but for any edge e∈E(G¯) and any color i∈[t], the addition of e to G in color i creates a rainbow copy of F. Let sat t(n,R(F)) denote the minimum number of edges in an (R(F),t)-saturated graph of order n. We call this the rainbow saturation number of F. In this article, we prove several results about rainbow saturation numbers of graphs. In stark contrast with the related problem for monochromatic subgraphs, wherein the saturation is always linear in n, we prove that rainbow saturation numbers have a variety of different orders of growth. For instance, the rainbow saturation number of the complete graph Kn lies between nlogn/loglogn and nlogn, the rainbow saturation number of an n-vertex star is quadratic in n, and the rainbow saturation number of any tree that is not a star is at most linear.

Uniqueness and minimal obstructions for tree-depth

A k -ranking of a graph G is a labeling of the vertices of G with values from { 1 , ? , k } such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for which a k -ranking of G exists. The graph G is k -critical if it has tree-depth k and every proper minor of G has smaller tree-depth.We establish partial results in support of two conjectures about the order and maximum degree of k -critical graphs. As part of these results, we define a graph G to be 1-unique if for every vertex v in G , there exists an optimal ranking of G in which v is the unique vertex with label 1. We show that several classes of k -critical graphs are 1-unique, and we conjecture that the property holds for all k -critical graphs. Generalizing a previously known construction for trees, we exhibit an inductive construction that uses 1-unique k -critical graphs to generate large classes of critical graphs having a given tree-depth.

Adjacency Relationships Forced by a Degree Sequence

There are typically several nonisomorphic graphs having a given degree sequence, and for any two degree sequence terms it is often possible to find a realization in which the corresponding vertices are adjacent and one in which they are not. We provide necessary and sufficient conditions for two vertices to be adjacent (or nonadjacent) in every realization of the degree sequence. These conditions generalize degree sequence and structural characterizations of the threshold graphs, in which every adjacency relationship is forcibly determined by the degree sequence. We further show that degree sequences for which adjacency relationships are forced form an upward-closed set in the dominance order on graphic partitions of an even integer.

On realization graphs of degree sequences

Given the degree sequence d of a graph, the realization graph of d is the graph having as its vertices the labeled realizations of d , with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I.?Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.

Non-minimal Degree-Sequence-Forcing Triples

Given a set