Graph partitions with minimum degree constraints

This paper gives some bounds for the vertex degree function index Hf(G) in terms of the order and size of a graph G, where G is a simple, finite and connected graph with minimum degree δ and maximum degree ∆. Some families of graphs are also constructed to show that the bounds can be achieved.

Partitions of graphs and multigraphs under degree constraints

By using the Szemerédi Regularity Lemma, Alon and Sudakov recently extended the classical Andrásfai-Erdős-Sós theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any $(r+1)$-partite graph $H$ whose smallest part has $t$ vertices, there exists a constant $C$ such that for any given $\varepsilon>0$ and sufficiently large $n$ the following is true. Whenever $G$ is an $n$-vertex graph with minimum degree $$\delta(G)\geq\left(1-{3\over 3r-1}+\varepsilon\right)n,$$ either $G$ contains $H$, or we can delete $f(n,H)\leq Cn^{2-{1\over t}}$ edges from $G$ to obtain an $r$-partite graph. Further, we are able to determine the correct order of magnitude of $f(n,H)$ in terms of the Zarankiewicz extremal function.

C. Thomassen and M. Szegedy proved the existence of a functionf(s, t) such that the points of anyf(s, t)-connected graph have a decomposition into two non-empty sets such that the subgraphs induced by them ares-connected andt-connected, respectively. We prove, thatf(s, t) ≦ 4s+4t − 13 and examine a similar problem for the minimum degree.

Structure of edges in plane graphs with bounded dual edge weight

The total domination number γ t ( G ) of a graph G is the order of a smallest set D ⊆ V ( G ) such that each vertex of G is adjacent to some vertex in D . The annihilation number a ( G ) of G is the largest integer k such that there exist k different vertices in G with degree sum of at most | E ( G )| . It is conjectured that γ t ( G ) ≤ a ( G ) + 1 holds for every nontrivial connected graph G . The conjecture was proved for graphs with minimum degree at least 3 , and remains unresolved for graphs with minimum degree 1 or 2 . In this paper we establish the conjecture for cactus graphs and block graphs.

The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques from data structures, graph algorithms, and scientific computing. In this paper, we present a simple but novel combinatorial algorithm for computing an exact minimum degree elimination ordering in $O(nm)$ time, which improves on the best known time complexity of $O(n^3)$ and offers practical improvements for sparse systems with small values of $m$. Our approach leverages a careful amortized analysis, which also allows us to derive output-sensitive bounds for the running time of $O(\min\{m\sqrt{m^+}, \Delta m^+\} \log n)$, where $m^+$ is the number of unique fill edges and original edges that the algorithm encounters and $\Delta$ is the maximum degree of the input graph. Furthermore, we show there cannot exist an exact minimum degree algorithm that runs in $O(nm^{1-\varepsilon})$ time, for any $\varepsilon > 0$, assuming the strong exponential time hypothesis. This fine-grained reduction goes through the orthogonal vectors problem and uses a new low-degree graph construction called $U$-fillers, which act as pathological inputs and cause any minimum degree algorithm to exhibit nearly worst-case performance. With these two results, we nearly characterize the time complexity of computing an exact minimum degree ordering.

For an edge-coloured graph G, the minimum colour degree of G means the minimum number of colours on edges which are incident to each vertex of G. We prove that if G is an edge-coloured graph with minimum colour degree at least 5, then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum colour degree at least 2. We show this theorem by proving amuch stronger form. Moreover, we point out an important relationship between our theorem and Bermond and Thomassen’s conjecture in digraphs.

We study the bipartite crossing number problem. When the minimum degree and the maximum degree of the graph are close to each other, we derive two polynomial time approximation algorithms for solving this problem, with approximation factors, O(log2 n), and O(log n log log n), from the optimal, respectively, where n is the number of vertices. This problem had been known to be NP-hard, but no approximation algorithm which could generate a provably good solution was known. An important aspect of our work has been relating this problem to the linear arrangement problem. Indeed using this relationship we also present an O(n 1.6) time algorithm for computing the bipartite crossing number of a tree.We also settle down the problem of computing a largest weighted biplanar subgraph of an acyclic graph by providing a linear time algorithm to it. This problem was known to be NP-hard when graph is planar and very sparse, and all weights are 1.KeywordsBipartite GraphLinear Time AlgorithmMedian PropertyLinear ArrangementPolynomial Time Approximation AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the existence of subgraphs with degree constraints

Directed graphs are widely used to model data flow and execution dependencies in streaming applications. This enables the utilization of graph partitioning algorithms for the problem of parallelizing execution on multiprocessor architectures under hardware resource constraints. However due to program memory restrictions in embedded multiprocessor systems, applications need to be divided into parts without cyclic dependencies. We found that this can be done by a subsequent second graph partitioning step with an additional acyclicity constraint. We have four main contributions. First, we show that this more constrained version of the graph partitioning problem is NP-complete and present linear time heuristics. We then integrate them into an existing multi-level graph partitioning framework to better handle large graphs. This achieves a 9% reduction of the edge cut compared to the previous single-level algorithm. Based on this, we engineer an evolutionary algorithm to further reduce the cut, achieving a 30% reduction on average compared to the state of the art. Finally, we integrate the partitioning heuristics into a graph compiler for an embedded multiprocessor architecture and show that this can reduce the amount of communication for a real-world imaging application and thereby accelerate it by an average of 11%. It is shown that the compiler can emit optimized code for vastly different hardware platforms using the heuristics. In addition, we demonstrate how a custom fitness function for the evolutionary algorithm can be used to optimize other objectives like load balancing if the communication volume is not predominantly important on a given hardware platform.

A graph is $k$-linked if for every list of $2k$ vertices $\{s_1,{\ldots}\,s_k, t_1,{\ldots}\,t_k\}$, there exist internally disjoint paths $P_1,{\ldots}\, P_k$ such that each $P_i$ is an $s_i,t_i$-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be $k$-linked.Let $D(n,k)$ be the minimum positive integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ is $k$-linked and let $R(n,k)$ be the minimum positive integer $r$ such that every $n$-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least $r$ is $k$-linked. The main result of the paper is finding the exact values of $D(n,k)$ and $R(n,k)$ for every $n$ and $k$.Thomas and Wollan [14] used the bound $D(n,k)\leq (n+3k)/2-2$ to give sufficient conditions for a graph to be $k$-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every $2k$-connected graph with average degree at least $12k$ is $k$-linked.

We investigate the structural and algorithmic properties of 2-community structures in graphs introduced recently by Olsen (Math Soc Sci 66(3):331–336, 2013). A 2-community structure is a partition of a vertex set into two parts such that for each vertex the numbers of neighbours in/outside its own part and the sizes of the parts are correlated. We show that some well studied graph classes as graphs of maximum degree 3, minimum degree at least |V|-3, trees and also others, have always a 2-community structure. Furthermore, a 2-community structure can be found in polynomial time in all these classes, even with additional request of connectivity in both parts. We introduce a concept of a weak 2-community and prove that in general graphs it is NP-complete to find a balanced weak 2-community structure with or without request for connectivity in both parts. On the other hand, we present a polynomial-time algorithm to solve the problem (without the condition for connectivity of parts) in graphs of degree at most 3.

On independent generalized degrees and independence numbers in K(1, m)-free graphs

Letk be a positive integer, and letG be a graph of ordern withn ? 4k ? 5,kn even and minimum degree at leastk. We prove that if the degree sum of each pair of nonadjacent vertices is at leastn, thenG has ak-factor.