Wiener Index and Remoteness in Triangulations and Quadrangulations

Let $G$ be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If $\overline{\sigma}(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the remoteness of $G$ is defined as the largest value of $\overline{\sigma}(v)$ over all vertices $v$ of $G$. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity

Inverse Problems Related to the Wiener and Steiner-Wiener Indices

In a graph, the generalized distance between multiple vertices is the minimum number of edges in a connected subgraph that contains these vertices. When we consider such distances between all subsets of $k$ vertices and take the sum, it is called the Steiner $k$-Wiener index and has important applications in Chemical Graph Theory. In this thesis we consider the inverse problems related to the Steiner Wiener index, i.e. for what positive integers is there a graph with Steiner Wiener index of that value

Streaming Complexity of Spanning Tree Computation

The semi-streaming model is a variant of the streaming model frequently used for the computation of graph problems. It allows the edges of an n-node input graph to be read sequentially in p passes using Õ(n) space. If the list of edges includes deletions, then the model is called the turnstile model; otherwise it is called the insertion-only model. In both models, some graph problems, such as spanning trees, k-connectivity, densest subgraph, degeneracy, cut-sparsifier, and (Δ+1)-coloring, can be exactly solved or (1+ε)-approximated in a single pass; while other graph problems, such as triangle detection and unweighted all-pairs shortest paths, are known to require Ω̃(n) passes to compute. For many fundamental graph problems, the tractability in these models is open. In this paper, we study the tractability of computing some standard spanning trees, including BFS, DFS, and maximum-leaf spanning trees. Our results, in both the insertion-only and the turnstile models, are as follows. Maximum-Leaf Spanning Trees: This problem is known to be APX-complete with inapproximability constant ρ ∈ [245/244, 2). By constructing an ε-MLST sparsifier, we show that for every constant ε > 0, MLST can be approximated in a single pass to within a factor of 1+ε w.h.p. (albeit in super-polynomial time for ε ≤ ρ-1 assuming P ≠ NP) and can be approximated in polynomial time in a single pass to within a factor of ρ_n+ε w.h.p., where ρ_n is the supremum constant that MLST cannot be approximated to within using polynomial time and Õ(n) space. In the insertion-only model, these algorithms can be deterministic. BFS Trees: It is known that BFS trees require ω(1) passes to compute, but the naïve approach needs O(n) passes. We devise a new randomized algorithm that reduces the pass complexity to O(√n), and it offers a smooth tradeoff between pass complexity and space usage. This gives a polynomial separation between single-source and all-pairs shortest paths for unweighted graphs. DFS Trees: It is unknown whether DFS trees require more than one pass. The current best algorithm by Khan and Mehta [STACS 2019] takes Õ(h) passes, where h is the height of computed DFS trees. Note that h can be as large as Ω(m/n) for n-node m-edge graphs. Our contribution is twofold. First, we provide a simple alternative proof of this result, via a new connection to sparse certificates for k-node-connectivity. Second, we present a randomized algorithm that reduces the pass complexity to O(√n), and it also offers a smooth tradeoff between pass complexity and space usage.ISSN:1868-896

Review of the New World genera of the Leafhopper Tribe Erythroneurini (Hemiptera: Cicadellidae: Typhlocycbinae)

The genus-level classification of New World Erythroneurini is revised based on results of a phylogenetic analysis of 100 morphological characters. The 704 known species are placed into 18 genera. Erasmoneura Young and Eratoneura Young, previously treated as subgenera of Erythroneura Fitch, and Erythridula Young, most recently treated as a subgenus of Arboridia Zachvatkin, are elevated to generic status. Three species previously included in Erasmoneura are placed in a new genus, Rossmoneura (type species, Erythroneura tecta McAtee). The concept of Erythroneura is thereby narrowed to include only those species previously included in the nominotypical subgenus. New World species previously included in Zygina Fieber are not closely related to the European type species of that genus and are therefore placed in new genera. Neozygina, n. gen., based on type species Erythroneura ceonothana Beamer, includes all species previously included in the ???ceonothana group???, and Zyginama, n. gen., based on type species Erythroneura ritana Beamer, includes most species previously included in the ???ritana group??? of New World Zygina. Five additional new genera are described to include other previously described North American Erythroneurini: Hepzygina, n. gen., based on type species Erythroneura milleri Beamer and also including E. aprica McAtee; Mexigina, n. gen., based on type species Erythroneura oculata McAtee; Nelionidia, n. gen., based on type species N. pueblensis, n. sp., three additional new species, and Erythroneura amicis Ross; Neoimbecilla, n. gen., based on type species Erythroneura kiperi Beamer and one new species; and Illinigina, n. gen., based on type species Erythroneura illinoiensis Gillette. Five new genera, based on previously undescribed species, are also recognized: Aztegina, n. gen, based on A. punctinota, n. sp., from Mexico; Amazygina, n. gen., based on type species A. decaspina, n. sp., and three additional new species from Ecuador; Hamagina, n. gen., based on type species H. spinigera, n. sp., and two additional new species from Peru and Ecuador; Napogina, n. gen., based on type species N. recta, n. sp., and one additional new species from Ecuador; Perugina, n. gen., based on type species P. denticula, n. sp., from Peru; and Spinigina, n. gen., based on type species S. hirsuta, n. sp., and an additional new species from Peru. Phylogenetic analysis suggests that the New World Erythroneurini consist of three lineages resulting from separate invasions from the Old World.published or submitted for publicationis peer reviewe

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version