Intersection representation of digraphs in trees with few leaves

The leafage of a digraph is the minimum number of leaves in a host tree in which it has a subtree intersection representation. We discuss bounds on the leafage in terms of other parameters (including Ferrers dimension), obtaining a string of sharp inequalities.Comment: 12 pages, 3 included figure

Iterative Delegations in Liquid Democracy with Restricted Preferences

In this paper, we study liquid democracy, a collective decision making paradigm which lies between direct and representative democracy. One main feature of liquid democracy is that voters can delegate their votes in a transitive manner so that: A delegates to B and B delegates to C leads to A delegates to C. Unfortunately, this process may not converge as there may not even exist a stable state (also called equilibrium). In this paper, we investigate the stability of the delegation process in liquid democracy when voters have restricted types of preference on the agent representing them (e.g., single-peaked preferences). We show that various natural structures of preferences guarantee the existence of an equilibrium and we obtain both tractability and hardness results for the problem of computing several equilibria with some desirable properties

On the Kernel and Related Problems in Interval Digraphs

Given a digraph $G$, a set $X\subseteq V(G)$ is said to be absorbing set (resp. dominating set) if every vertex in the graph is either in $X$ or is an in-neighbour (resp. out-neighbour) of a vertex in $X$. A set $S\subseteq V(G)$ is said to be an independent set if no two vertices in $S$ are adjacent in $G$. A kernel (resp. solution) of $G$ is an independent and absorbing (resp. dominating) set in $G$. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph $G$ is an interval digraph if a pair of intervals $(S_u,T_u)$ can be assigned to each vertex $u$ of $G$ such that $(u,v)\in E(G)$ if and only if $S_u\cap T_v\neq\emptyset$. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs -- which arise when we require that the two intervals assigned to a vertex have to intersect. We show that all the problems mentioned above are efficiently solvable, in most of the cases even linear-time solvable, in the class of reflexive interval digraphs, but are APX-hard on even the very restricted class of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate, i.e. they consist of a single point each. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs.Comment: 26 pages, 3 figure

The Distribution of the Domination Number of a Family of Random Interval Catch Digraphs

We study a new kind of proximity graphs called proportional-edge proximity catch digraphs (PCDs)in a randomized setting. PCDs are a special kind of random catch digraphs that have been developed recently and have applications in statistical pattern classification and spatial point pattern analysis. PCDs are also a special type of intersection digraphs; and for one-dimensional data, the proportional-edge PCD family is also a family of random interval catch digraphs. We present the exact (and asymptotic) distribution of the domination number of this PCD family for uniform (and non-uniform) data in one dimension. We also provide several extensions of this random catch digraph by relaxing the expansion and centrality parameters, thereby determine the parameters for which the asymptotic distribution is non-degenerate. We observe sudden jumps (from degeneracy to non-degeneracy or from a non-degenerate distribution to another) in the asymptotic distribution of the domination number at certain parameter values.Comment: 29 pages, 3 figure

Classes of Intersection Digraphs with Good Algorithmic Properties

While intersection graphs play a central role in the algorithmic analysis of hard problems on undirected graphs, the role of intersection digraphs in algorithms is much less understood. We present several contributions towards a better understanding of the algorithmic treatment of intersection digraphs. First, we introduce natural classes of intersection digraphs that generalize several classes studied in the literature. Second, we define the directed locally checkable vertex (DLCV) problems, which capture many well-studied problems on digraphs such as (Independent) Dominating Set, Kernel, and H-Homomorphism. Third, we give a new width measure of digraphs, bi-mim-width, and show that the DLCV problems are polynomial-time solvable when we are provided a decomposition of small bi-mim-width. Fourth, we show that several classes of intersection digraphs have bounded bi-mim-width, implying that we can solve all DLCV problems on these classes in polynomial time given an intersection representation of the input digraph. We identify reflexivity as a useful condition to obtain intersection digraph classes of bounded bi-mim-width, and therefore to obtain positive algorithmic results

Distribution of the Relative Density of Central Similarity Proximity Catch Digraphs Based on One Dimensional Uniform Data

We consider the distribution of a graph invariant of central similarity proximity catch digraphs (PCDs) based on one dimensional data. The central similarity PCDs are also a special type of parameterized random digraph family defined with two parameters, a centrality parameter and an expansion parameter, and for one dimensional data, central similarity PCDs can also be viewed as a type of interval catch digraphs. The graph invariant we consider is the relative density of central similarity PCDs. We prove that relative density of central similarity PCDs is a U-statistic and obtain the asymptotic normality under mild regularity conditions using the central limit theory of U-statistics. For one dimensional uniform data, we provide the asymptotic distribution of the relative density of the central similarity PCDs for the entire ranges of centrality and expansion parameters. Consequently, we determine the optimal parameter values at which the rate of convergence (to normality) is fastest. We also provide the connection with class cover catch digraphs and the extension of central similarity PCDs to higher dimensions.Comment: 28 pages, 6 figure

On some subclasses of circular-arc catch digraphs

Catch digraphs was introduced by Hiroshi Maehera in 1984 as an analog of intersection graphs where a family of pointed sets represents a digraph. After that Prisner continued his research particularly on interval catch digraphs by characterizing them diasteroidal triple free. It has numerous applications in the field of real world problems like network technology and telecommunication operations. In this article we introduce a new class of catch digraphs, namely circular-arc catch digraphs. The definition is same as interval catch digraph, only the intervals are replaced by circular-arcs here. We present the characterization of proper circular-arc catch digraphs, which is a natural subclass of circular-arc catch digraphs where no circular-arc is contained in other properly. We do the characterization by introducing a concept monotone circular ordering for the vertices of the augmented adjacency matrices of it. Next we find that underlying graph of a proper oriented circular-arc catch digraph is a proper circular-arc graph. Also we characterize proper oriented circular-arc catch digraphs by defining a certain kind of circular vertex ordering of its vertices. Another interesting result is to characterize oriented circular-arc catch digraphs which are tournaments in terms of forbidden subdigraphs. Further we study some properties of an oriented circular-arc catch digraph. In conclusion we discuss the relations between these subclasses of circular-arc catch digraphs