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.Comment: 18 pages, 4 figure

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 Hammer et al. and Merris, 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.Comment: 15 pages, 6 figure

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.Comment: 14 pages, 4 figure

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.Comment: 10 pages, 5 figure

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, Mah\'eo, and Sacl\'e, 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.Comment: 7 pages, 3 figure

Bat avoidance behavior in Neoconocephalus retusus (Orthoptera: Tettigoniidae) [abstract]

Abstract only availableFaculty Mentor: Dr. Johannes Schul, BiologyNocturnally flying insects avoid predation by insectivorous bats by monitoring for echolocation calls and, when detected, using evasive behaviors. We examined the responses of Neoconocephalus retusus to models of bat calls during tethered flight. The bat calls were 8 ms in duration and frequency modulated (FM) from 80 to 35 kHz with the main energy concentrated at 40 kHz. Such calls resemble the search phase calls of the Big Brown Bat (Eptesicus fuscus), an insectivorous bat commonly found in Missouri. We simulated a bat in the search phase as if it were approaching the katydid to characterize the response types of N. retusus. At low amplitudes of the bat call, equivalent to a distance of approximately 24m from the insect, N. retusus steered away from the sound source by extending one hind leg. At higher intensities (~20m from the insect) the insect interrupted the beating of the hind wings while remaining in flight posture. At even higher bat call intensities, corresponding to approximately 15m from the insect, flight stopped and the animal assumed a dive position: both sets of wings folded up, the antennae folded back, and all legs were placed flush against the abdomen. All three behaviors occur before a bat would be able to hear the katydid's echo. We used a series of 7 FM pulses of constant amplitude to determine the thresholds for the three behaviors. The thresholds for steering, interruption of flight and cessation of flight are 46.5 dB SPL, 51 dB SPL, and 59.5 dB SPL respectively. We also examined the thresholds using a single FM pulse in which the thresholds were approximately 5-6 dB higher. Bat avoidance in N. retusus differs strikingly from that of the closely related N. ensiger, which shows only a behavior equivalent to our interruption of wingbeat response, but no steering or dive behaviors. However, the avoidance behaviors of N. retusus were similar to those of a more distantly related katydid as well as those of crickets and some moths. Thus, bat evasive behavior might differ significantly among closely related species, but often shows convergence among distantly related groups. This suggests that such behaviors readily adapt to changing predation pressures rather than being conserved within phylogenic groups.Life Sciences Undergraduate Research Opportunity Progra