Paradigms for Effective Parallelization of Inherently Sequential Graph Algorithms on Multi-Core Architectures

The chapter describes two algorithmic paradigms, dubbed speculation and iteration and approximate update, for parallelizing greedy graph algorithms and vertex ordering algorithms, respectively, on multicore architectures. The common challenge in these two classes of algorithms is that the computations involved are inherently sequential. The efficacy of the paradigms in overcoming this challenge is demonstrated via extensive experimental study on two representative algorithms from each class and two Intel multi-core systems. The algorithms studied are (1) greedy algorithms for distance-k coloring (for k = 1 and k = 2) and (2) algorithms for two degree-based vertex orderings. The experimental results show that the paradigms enable the design of scalable methods that to a large extent preserve the quality of solution obtained by the underlying serial algorithms.

On the Minimum Size of Hamilton Saturated Hypergraphs

For $1\leq \ell< k$, an $\ell$-overlapping $k$-cycle is a $k$-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two consecutive edges share exactly $\ell$ vertices. A $k$-uniform hypergraph $H$ is $\ell$-hamiltonian saturated if $H$ does not contain an $\ell$-overlapping hamiltonian $k$-cycle but every hypergraph obtained from $H$ by adding one edge does contain such a cycle. Let sat$(N,k,\ell)$ be the smallest number of edges in an $\ell$-hamiltonian saturated $k$-uniform hypergraph on $N$ vertices. In the case of graphs Clark and Entringer showed in 1983 that sat$(N,2,1)=\lceil \tfrac{3N}2\rceil$. The present authors proved that for $k\geq 3$ and $\ell=1$, as well as for all $0.8k\leq \ell\leq k-1$, sat$(N,k,\ell)=\Theta(N^{\ell})$. Here we prove that sat$(N,2\ell,\ell)=\Theta\left(N^\ell\right)$.

Coloring squares of graphs via vertex orderings

We provide upper bounds on the chromatic number of the square of graphs, which have vertex ordering characterizations. We prove that [Formula: see text] is [Formula: see text]-colorable when [Formula: see text] is a cocomparability graph, [Formula: see text]-colorable when [Formula: see text] is a strongly orderable graph and [Formula: see text]-colorable when [Formula: see text] is a dually chordal graph, where [Formula: see text] is the maximum degree and [Formula: see text] = max[Formula: see text] is the multiplicity of the graph [Formula: see text]. This improves the currently known upper bounds on the chromatic number of squares of graphs from these classes.

On Grounded L-Graphs and their Relatives

We consider the graph classes Grounded-L and Grounded-{𝖫,⅃} corresponding to graphs that admit an intersection representation by 𝖫-shaped curves (or 𝖫-shaped and ⅃-shaped curves, respectively), where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns. We also compare these classes to related intersection classes, such as the grounded segment graphs, the monotone 𝖫-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinct and satisfy only trivial or previously known inclusions.