arXiv subjects

Michael D. Barrus

Publications and source records attributed to Michael D. Barrus.

At least 19 recordsLinked to original sources

arXiv2026

The forbidden structure for zero forcing number

The {\it zero forcing number} of a graph $G$, $Z(G)$, is a well-studied parameter which arises from a color changing process and has strong connections to {\it minimum rank}, {\it critical ideals} and related invariants. In this work, we consider the complementary parameter $\mz(G) = |V(G)| - Z(G)$. This parameter is monotone under taking induced subgraphs. This leads us to the study of graphs for which $\mz(G)$ is bounded, via forbidden induced subgraphs. We prove that the number of minimal forbidden graphs for graphs with $\mz(G)\leq k$ is finite for any $k\geq 1$. We determine the complete set of minimal forbidden graphs for the case $k = 3$, and we provide partial characterizations of graphs with $\mz(G) \leq 3$, based on girth. Our results suggest new directions for the structural understanding of zero forcing-type parameters.

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arXiv2025

A characterization of all graphs cospectral to the double star $P_2(1,n)$

We examine the adjacency spectrum of trees with diameter three, also referred to as double stars. Using $P_2(a,b)$ to denote a double star with $ a$ and $b$ leaves at its respective endpoints, we discuss graphs which are cospectral to double stars for various parameters $a$ and $b$. In particular, we give constructions for graphs cospectral to $P_2(1,2k)$ for integers $k$. Lastly, we show that the double star $P_2(1,n)$ is determined by its spectrum when $n$ is odd. That is, if a graph $G$ cospectral to $P_2(1,n)$ for odd $n$, then $G$ is isomorphic to $P_2(1,n)$.

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arXiv2023

The Hereditary Closure of the Unigraphs

A graph with degree sequence $π$ is a \emph{unigraph} if it is isomorphic to every graph that has degree sequence $π$. The class of unigraphs is not hereditary and in this paper we study the related hereditary class HCU, the hereditary closure of unigraphs, consisting of all graphs induced in a unigraph. We characterize the class HCU in multiple ways making use of the tools of a decomposition due to Tyshkevich and a partial order on degree sequences due to Rao. We also provide a new characterization of the class that consists of unigraphs for which all induced subgraphs are also unigraphs.

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arXiv2023

Distinguishing chromatic number of Hamiltonian circulant graphs

The distinguishing chromatic number of a graph $G$ is the smallest number of colors needed to properly color the vertices of $G$ so that the trivial automorphism is the only symmetry of $G$ that preserves the coloring. We investigate the distinguishing chromatic number for Hamiltonian circulant graphs with maximum degree at most 4.

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arXiv2022

Cliques in realization graphs

The realization graph $\mathcal{G}(d)$ of a degree sequence $d$ is the graph whose vertices are labeled realizations of $d$, where edges join realizations that differ by swapping a single pair of edges. Barrus [On realization graphs of degree sequences, Discrete Mathematics, vol. 339 (2016), no. 8, pp. 2146-2152] characterized $d$ for which $\mathcal{G}(d)$ is triangle-free. Here, for any $n \geq 4$, we describe a structure in realizations of $d$ that exactly determines whether $G(d)$ has a clique of size $n$. As a consequence we determine the degree sequences $d$ for which $\mathcal{G}(d)$ is a complete graph on $n$ vertices.

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arXiv2022

Upward-closed hereditary families in the dominance order

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris (2002), the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of $e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes $e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3.

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arXiv2021

The principal Erdős--Gallai differences of a degree sequence

The Erdős--Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erdős--Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao's order on degree sequences.

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arXiv2020

Graphs with few trivial characteristic ideals

We give a characterization of the graphs with at most three trivial characteristic ideals. This implies the complete characterization of the regular graphs whose critical groups have at most three invariant factors equal to 1 and the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1. We also give an alternative and simpler way to obtain the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1, and a list of minimal forbidden graphs for the family of graphs with Smith group having at most 4 invariant factors equal to 1.

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arXiv2018

Weakly threshold graphs

We define a weakly threshold sequence to be a degree sequence $d=(d_1,\dots,d_n)$ of a graph having the property that $\sum_{i \leq k} d_i \geq k(k-1)+\sum_{i > k} \min\{k,d_i\} - 1$ for all positive $k \leq \max\{i:d_i \geq i-1\}$. The weakly threshold graphs are the realizations of the weakly threshold sequences. The weakly threshold graphs properly include the threshold graphs and satisfy pleasing extensions of many properties of threshold graphs. We demonstrate a majorization property of weakly threshold sequences and an iterative construction algorithm for weakly threshold graphs, as well as a forbidden induced subgraph characterization. We conclude by exactly enumerating weakly threshold sequences and graphs.

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arXiv2017

On 1-uniqueness and dense critical graphs for tree-depth

The tree-depth of $G$ is the smallest value of $k$ for which a labeling of the vertices of $G$ with elements from $\{1,\dots,k\}$ exists such that any path joining two vertices with the same label contains a vertex having a higher label. The graph $G$ is $k$-critical if it has tree-depth $k$ and every proper minor of $G$ has smaller tree-depth. Motivated by a conjecture on the maximum degree of $k$-critical graphs, we consider the property of 1-uniqueness, wherein any vertex of a critical graph can be the unique vertex receiving label 1 in an optimal labeling. Contrary to an earlier conjecture, we construct examples of critical graphs that are not 1-unique and show that 1-unique graphs can have arbitrarily many more edges than certain critical spanning subgraphs. We also show that $(n-1)$-critical graphs are 1-unique and use 1-uniqueness to show that the Andrásfai graphs are critical with respect to tree-depth.

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arXiv2016

Minimal obstructions for tree-depth: A non-1-unique example

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. As defined in [M. D. Barrus, J. Sinkovic, Uniqueness and minimal obstructions for tree-depth, Discrete Math 339 (2) (2015) 606-613], a graph G is 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. In the above paper and [M. D. Barrus, J. Sinkovic, Classes of critical graphs for tree-depth, arXiv:1502.0577] the authors showed that several classes of critical graphs are 1-unique and asked whether all critical graphs have this property. We answer in the negative by demonstrating an infinite family of graphs which are critical but not 1-unique.

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arXiv2016

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.

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arXiv2015

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.

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arXiv2015

Neighborhood degree lists of graphs

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.

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arXiv2015

Graphs with the strong Havel-Hakimi property

The Havel-Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel-Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel-Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set.

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arXiv2015

Classes of critical graphs 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 any proper minor of G has smaller tree-depth, and it is 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 present several classes of graphs that are both k-critical and 1-unique, providing examples that satisfy conjectures on critical graphs discussed in [M.D. Barrus and J. Sinkovic, Uniqueness and minimal obstructions for tree-depth, submitted].

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arXiv2015

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.

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arXiv2013

On fractional realizations of graph degree sequences

We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope. We describe properties of the polytope vertices and characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of pseudo-split graphs, and we characterize their realizations both in terms of forbidden subgraphs and graph structure.

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