16,929 research outputs found
Spanning Trees of Bounded Degree Graphs
We consider lower bounds on the number of spanning trees of connected graphs
with degree bounded by . The question is of interest because such bounds may
improve the analysis of the improvement produced by memorisation in the runtime
of exponential algorithms. The value of interest is the constant such
that all connected graphs with degree bounded by have at least
spanning trees where is the cyclomatic number or excess of
the graph, namely . We conjecture that is achieved by the
complete graph but we have not proved this for any greater than
3. We give weaker lower bounds on for
The bandwidth theorem in sparse graphs
The bandwidth theorem [Mathematische Annalen, 343(1):175â205, 2009] states that any n-vertex graph G with minimum degree [Formula Presented] contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for p â«[Formula Presented] asymptotically almost surely each spanning subgraph G of G(n, p) with minimum degree [Formula Presented] pn contains all n-vertex k-colourable graphs H with maximum degree â, bandwidth o(n), and at least Cpâ2 vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of spanning subgraphs. Finally, we provide improved results for H with small degeneracy, which in particular imply a resilience result in G(n, p) with respect to the containment of spanning bounded degree trees for p â«[Formula Presented]
Minimum vertex degree conditions for loose spanning trees in 3-graphs
In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large
-vertex graph with minimum degree at least contains all
spanning trees of bounded degree. We consider a generalization of this result
to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained
by successively appending edges sharing a single vertex with a previous edge.
We show that for all and , and large, every -vertex
3-uniform hypergraph of minimum vertex degree
contains every loose spanning tree with maximum vertex degree .
This bound is asymptotically tight, since some loose trees contain perfect
matchings.Comment: 18 pages, 1 figur
Thin Trees in Some Families of Graphs
Let =(,) be a graph and let be a spanning tree of . The thinness parameter of denoted by () is the maximum over all cuts of the proportion of the edges of in the cut. Thin trees play an important role in some recent papers on the Asymmetric Traveling Salesman Problem (ATSP). Goddyn conjectured that every graph of sufficiently large edge-connectivity has a spanning tree such that () †.
In this thesis, we study the problem of finding thin spanning trees in two families of graphs, namely, (1) distance-regular graphs (DRGs), and (2) planar graphs.
For some families of DRGs such as strongly regular graphs, Johnson graphs, Crown graphs, and Hamming graphs, we give a polynomial-time construction of spanning trees of maximum degree †3 such that () is determined by the parameters of the graph.
For planar graphs, we improve the analysis of Merker and Postle ("Bounded Diameter Arboricity", arXiv:1608.05352v1) and show that every 6-edge-connected planar graph has two edge-disjoint spanning trees ,âČ such that (),(âČ) †14â15. For 8-edge-connected planar graphs , we present a simplified version of the techniques of Merker and Postle and show that has two edge-disjoint spanning trees ,âČ such that (),(âČ) †12â13
A Local Algorithm for the Sparse Spanning Graph Problem
Constructing a sparse spanning subgraph is a fundamental primitive in graph
theory. In this paper, we study this problem in the Centralized Local model,
where the goal is to decide whether an edge is part of the spanning subgraph by
examining only a small part of the input; yet, answers must be globally
consistent and independent of prior queries.
Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees,
cannot be constructed efficiently in this model. Therefore, we settle for a
spanning subgraph containing at most edges (where is the
number of vertices and is a given approximation/sparsity
parameter). We achieve query complexity of
, (-notation hides
polylogarithmic factors in ). where is the maximum degree of the
input graph. Our algorithm is the first to do so on arbitrary bounded degree
graphs. Moreover, we achieve the additional property that our algorithm outputs
a spanner, i.e., distances are approximately preserved. With high probability,
for each deleted edge there is a path of
hops in the output that connects its endpoints
A Centralized Local Algorithm for the Sparse Spanning Graph Problem
Constructing a sparse spanning subgraph is a fundamental primitive in graph theory. In this paper, we study this problem in the Centralized Local model, where the goal is to decide whether an edge is part of the spanning subgraph by examining only a small part of the input; yet, answers must be globally consistent and independent of prior queries.
Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees, cannot be constructed efficiently in this model. Therefore, we settle for a spanning subgraph containing at most (1+epsilon)n edges (where n is the number of vertices and epsilon is a given approximation/sparsity parameter). We achieve a query complexity of O~(poly(Delta/epsilon)n^{2/3}), where Delta is the maximum degree of the input graph. Our algorithm is the first to do so on arbitrary bounded degree graphs. Moreover, we achieve the additional property that our algorithm outputs a spanning subgraph of bounded stretch i.e., distances are approximately preserved. With high probability, for each deleted edge there is a path of O(log n * (Delta+log n)/epsilon) hops in the output that connects its endpoints
A Fast Graph Program for Computing Minimum Spanning Trees
When using graph transformation rules to implement graph algorithms, a
challenge is to match the efficiency of programs in conventional languages. To
help overcome that challenge, the graph programming language GP 2 features
rooted rules which, under mild conditions, can match in constant time on
bounded degree graphs. In this paper, we present an efficient GP 2 program for
computing minimum spanning trees. We provide empirical performance results as
evidence for the program's subquadratic complexity on bounded degree graphs.
This is achieved using depth-first search as well as rooted graph
transformation. The program is based on Boruvka's algorithm for minimum
spanning trees. Our performance results show that the program's time complexity
is consistent with that of classical implementations of Boruvka's algorithm,
namely O(m log n), where m is the number of edges and n the number of nodes.Comment: In Proceedings GCM 2020, arXiv:2012.0118
Engineering Fused Lasso Solvers on Trees
The graph fused lasso optimization problem seeks, for a given input signal y=(y_i) on nodes i? V of a graph G=(V,E), a reconstructed signal x=(x_i) that is both element-wise close to y in quadratic error and also has bounded total variation (sum of absolute differences across edges), thereby favoring regionally constant solutions. An important application is denoising of spatially correlated data, especially for medical images.
Currently, fused lasso solvers for general graph input reduce the problem to an iteration over a series of "one-dimensional" problems (on paths or line graphs), which can be solved in linear time. Recently, a direct fused lasso algorithm for tree graphs has been presented, but no implementation of it appears to be available.
We here present a simplified exact algorithm and additionally a fast approximation scheme for trees, together with engineered implementations for both. We empirically evaluate their performance on different kinds of trees with distinct degree distributions (simulated trees; spanning trees of road networks, grid graphs of images, social networks). The exact algorithm is very efficient on trees with low node degrees, which covers many naturally arising graphs, while the approximation scheme can perform better on trees with several higher-degree nodes when limiting the desired accuracy to values that are useful in practice
Universality for bounded degree spanning trees in randomly perturbed graphs
We solve a problem of Krivelevich, Kwan and Sudakov concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph G α on n vertices with ÎŽ(G α ) ℠αn for α > 0 and we add to it the binomial random graph G(n,C/n), then with high probability the graph G α âȘG(n,C/n) contains copies of all spanning trees with maximum degree at most Î simultaneously, where C depends only on α and Î
- âŠ