We consider the graph classes Grounded-L and Grounded-{L, L} corresponding to graphs that admit an intersection representation by L-shaped curves (or L-shaped and L-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.
Biró, Hujter, and Tuza (1992) introduced the concept of H-graphs, intersection graphs of connected subregions of a graph H thought of as a one-dimensional topological space. They are related to and generalize many important classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. Our paper starts a new line of research in the area of geometric intersection graphs by studying H-graphs from the point of view of computational problems that are fundamental in theoretical computer science: recognition, graph isomorphism, dominating set, clique, and colorability.Surprisingly, we negatively answer the 25-year-old question of Biró, Hujter, and Tuza which asks whether H-graphs can be recognized in polynomial time, for a fixed graph H. We prove that it is NP-complete if H contains the diamond graph as a minor. On the positive side, we provide a polynomial-time algorithm recognizing T -graphs, for each fixed tree T . For the special case when T is a star S d of degree d, we have an O(n 3.5 )-time algorithm.We give FPTand XP-time algorithms solving the minimum dominating set problem on S dgraphs and H-graphs parametrized by d and the size of H, respectively. As a byproduct, the algorithm for H-graphs adapts to an XP-time algorithm for the independent set and the independent dominating set problems on H-graphs.If H contains the double-triangle as a minor, we prove that the graph isomorphism problem is GI-complete and that the clique problem is APX-hard. On the positive side, we show that the clique problem can be solved in polynomial time if H is a cactus graph. Also, when a graph G has a Helly H-representation, the clique problem can be solved in polynomial time.Further, we show that both the k-clique and the list k-coloring problems are solvable in FPTtime on H-graphs (parameterized by k and the treewidth of H). In fact, these results apply to classes graphs of graphs with treewidth bounded by a function of the clique number.Finally, we observe that H-graphs have at most n O( H ) minimal separators which allows us to apply the meta-algorithmic framework of Fomin, Todinca, and Villanger (2015) to show that for each fixed t, finding a maximum induced subgraph of treewidth t can be done in polynomial time. In the case when H is a cactus, we improve the bound to O( H n 2 ).
No abstract
There has recently been a surge of interest in the computational and complexity properties of the population model, which assumes anonymous, computationally-bounded nodes, interacting at random, with the goal of jointly computing global predicates. Significant work has gone towards investigating majority or consensus dynamics in this model: that is, assuming that every node is initially in one of two states or , determine which state had higher initial count. In this paper, we consider a natural generalization of majority/consensus, which we call comparison: in its simplest formulation, we are given two baseline states, 0 and 0 , present in any initial configuration in fixed, but possibly small counts. One of these states has higher count than the other: we will assume | 0 | ≥ | 0 | for some constant > 1. The challenge is to design a protocol by which nodes can quickly and reliably decide on which of the baseline states 0 and 0 has higher initial count.We begin by analyzing a simple and general dynamics solving the above comparison problem, which uses O (log ) states per node, and converges in O (log ) (parallel) time, with high probability, to a state where the whole population votes on opinions or at rates proportional to the initial concentrations of | 0 | vs. | 0 |. We then describe how this procedure can be bootstrapped to solve comparison, i.e. have every node in the population reach the "correct" decision, with probability 1 − (1), at the cost of O (log log ) additional states. Further, we prove that this dynamics is self-stabilizing, in the sense that it converges to the correct decision from arbitrary initial states, and leak-robust, in the sense that it can withstand spurious faulty reactions, which are known to occur in practical implementations of population protocols. Our analysis is based on a new martingale concentration result relating the discrete-time evolution of a population protocol to its expected (steady-state) analysis, which should be a useful tool when analyzing opinion dynamics and epidemic dissemination in the population model.
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