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Register a new userOn the Connectedness and Diameter of a Geometric Johnson Graph
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Added by
Ruy Fabila-Monroy
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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Let $P$ be a set of $n$ points in general position in the plane. A subset $I$ of $P$ is called an emph{island} if there exists a convex set $C$ such that $I = P cap C$. In this paper we define the emph{generalized island Johnson graph} of $P$ as the graph whose vertex consists of all islands of $P$ of cardinality $k$, two of which are adjacent if their intersection consists of exactly $l$ elements. We show that for large enough values of $n$, this graph is connected, and give upper and lower bounds on its diameter.
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Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length $n$, the classical associahedra, is $2n-6$ for a large enough $n$. We give a tight bound of $Theta(m)$ on the diameter of trivially perfect graph associahedra on $m$ edges. We consider the maximum diameter of associahedra of graphs on $n$ vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is $Theta (nlog n)$.
We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for integer polytopes in terms of the length of the description of the polytope (in bits) and the minimum angle between facets of its polar. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a giant component of vertices, with measure $1-o(1)$ and polynomial diameter. Both bounds rely on spectral gaps -- of a certain Schrodinger operator in the first case, and a certain continuous time Markov chain in the second -- which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.
A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $min {{rm deg}_G(x), {rm deg}_G(y)}$ vertex(edge)-disjoint paths between $x$ and $y$. In this paper, we consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube $AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ for $ngeq 4$ (resp. $AQ_{n,k}$ for $ngeq 2$ and $kgeq 4$) is still strongly Menger connected even when there are $4n-9$ (resp. $4n-8$) faulty vertices and $AQ_{n,k}$ is still strongly Menger edge connected even when there are $4n-4$ faulty edges for $ngeq 2$ and $kgeq 3$. Moreover, under the restricted condition that each vertex has at least two fault-free edges, we show that $AQ_{n,k}$ is still strongly Menger edge connected even when there are $8n-10$ faulty edges for $ngeq 2$ and $kgeq 3$. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.
Graham and Pollak showed that the vertices of any graph $G$ can be addressed with $N$-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length $N$ is minimum possible. In this paper, we determine an addressing of length $k(n-k)$ for the Johnson graphs $J(n,k)$ and we show that our addressing is optimal when $k=1$ or when $k=2, n=4,5,6$, but not when $n=6$ and $k=3$. We study the addressing problem as well as a variation of it in which the alphabet used has more than three symbols, for other graphs such as complete multipartite graphs and odd cycles. We also present computations describing the distribution of the minimum length of addressings for connected graphs with up to $10$ vertices. Motivated by these computations we settle a problem of Graham, showing that most graphs on $n$ vertices have an addressing of length at most $n-(2-o(1))log_2 n$.
The behavior of a certain random growth process is analyzed on arbitrary regular and non-regular graphs. Our argument is based on the Expander Mixing Lemma, which entails that the results are strongest for Ramanujan graphs, which asymptotically maximize the spectral gap. Further, we consider ErdH{o}s--Renyi random graphs and compare our theoretical results with computational experiments on flip graphs of point configurations. The latter is relevant for enumerating triangulations.
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