Light Spanners

A $t$-spanner of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality -- in this work we focus on the latter. Specifically, it is shown that for any parameters $k\ge 1$ and $\epsilon>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+\epsilon)$-stretch spanner of weight at most $w(MST(G))\cdot O_\epsilon(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a minimum spanning tree of $G$. Our result is obtained via a novel analysis of the classic greedy algorithm, and improves previous work by a factor of $O(\log k)$.Comment: 10 pages, 1 figure, to appear in ICALP 201

Spanning Properties of Theta-Theta Graphs

We study the spanning properties of Theta-Theta graphs. Similar in spirit with the Yao-Yao graphs, Theta-Theta graphs partition the space around each vertex into a set of k cones, for some fixed integer k > 1, and select at most one edge per cone. The difference is in the way edges are selected. Yao-Yao graphs select an edge of minimum length, whereas Theta-Theta graphs select an edge of minimum orthogonal projection onto the cone bisector. It has been established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio 11.67, for k' >= 6. In this paper we establish a first spanning ratio of $7.82$ for Theta-Theta graphs, for the same values of $k$. We also extend the class of Theta-Theta spanners with parameter 6k', and establish a spanning ratio of $16.76$ for k' >= 5. We surmise that these stronger results are mainly due to a tighter analysis in this paper, rather than Theta-Theta being superior to Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta graphs decreases to 4.64 as k' increases to 8. These are the first results on the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table

On a Reconstruction Problem for Sequences

AbstractIt is shown that any word of lengthnis uniquely determined by all its[formula]subwords of lengthk, providedk⩾⌊167n⌋+5. This improves the boundk⩾⌊n/2⌋ given in B. Manvelet al.(Discrete Math.94(1991), 209–219)

Switching reconstruction and diophantine equations

AbstractBased on a result of R. P. Stanley (J. Combin. Theory Ser. B 38, 1985, 132–138) we show that for each s ≥ 4 there exists an integer Ns such that any graph with n > Ns vertices is reconstructible from the multiset of graphs obtained by switching of vertex subsets with s vertices, provided n ≠ 0 (mod 4) if s is odd. We also establish an analog of P. J. Kelly's lemma (Pacific J. Math., 1957, 961–968) for the above s-switching reconstruction problem

More on Vertex-Switching Reconstruction

AbstractA graph is called s-vertex switching reconstructible (s-VSR) if it is uniquely defined, up to isomorphism by the multiset of unlabeled graphs obtained by switching of all its s-vertex subsets. Stanley proved that a graph with n vertices is s-VSR if the Krawtchouk polynomial Pns has no even roots. Solving balance equations, introduced in Krasnikov and Roditty (Arch. Math. (Basel) 48 (1987). 458-464) for the switching reconstruction problem, we show that a graph is s-VSR if the corresponding Krawtchouk polynomial has one or two even roots laying far from n/2. As a consequence we prove that graphs with sufficiently large number n of vertices are s-VSR for some values of s about n/2. In particular, all graphs are s-VSR for n − 2s = 0, 1, 3. and if n ≠ 0 (mod 4), for n − 2s = 2, 6

Zero-sum partition theorems for graphs

Let q=pn be a power of an odd prime p. We show that the vertices of every graph G can be partitioned into t(q) classes V(G)=⋃t=1t(q)Vi such that the number of edges in any induced subgraph 〈Vi〉 is divisible by q, where t(q)≤32(q−1)−(2(q−1)−1)124+98, and if q=2n, then t(q)=2q−1

Reconstructing graphs from their k-edge deleted subgraphs

AbstractLet G be a graph with m edges and n vertices. We show that if 2m−k>n! or if 2m>(2n) + k then G is determined by its collection of k-edge deleted subgraphs

On induced subgraphs of trees, with restricted degrees

AbstractIt is proved that every tree T on n⩾2 vertices contains an induced subgraph F such that all its degrees are odd and |F|⩾⌈n2⌉

On the largest tree of given maximum degree in a connected graph

We prove that every connected graph G contains a tree T of maximum degree at most k that either spans G or has order at least k(G) + 1, where (G) is the minimum degree of G. This generalizes and unifies earlier results of Bermond [1] and Win [7]. We also show that the square of a connected graph contains a spanning tree of maximum degree at most three