Maximal planar networks with large clustering coefficient and power-law degree distribution

In this article, we propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called {\bf Random Apollonian Networks}(RAN) as they can be considered as a variation of Apollonian networks. We obtain the analytic results of power-law exponent $\gamma =3$ and clustering coefficient $C={46/3}-36\texttt{ln}{3/2}\approx 0.74$, which agree very well with the simulation results. We prove that the increasing tendency of average distance of RAN is a little slower than the logarithm of the number of nodes in RAN. Since most real-life networks are both scale-free and small-world networks, RAN may perform well in mimicking the reality. The RAN possess hierarchical structure as $C(k)\sim k^{-1}$ that in accord with the observations of many real-life networks. In addition, we prove that RAN are maximal planar networks, which are of particular practicability for layout of printed circuits and so on. The percolation and epidemic spreading process are also studies and the comparison between RAN and Barab\'{a}si-Albert(BA) as well as Newman-Watts(NW) networks are shown. We find that, when the network order $N$(the total number of nodes) is relatively small(as $N\sim 10^4$), the performance of RAN under intentional attack is not sensitive to $N$, while that of BA networks is much affected by $N$. And the diseases spread slower in RAN than BA networks during the outbreaks, indicating that the large clustering coefficient may slower the spreading velocity especially in the outbreaks.Comment: 13 pages, 10 figure

Broadcast dimension of graphs

In this paper we initiate the study of broadcast dimension, a variant of metric dimension. Let G be a graph with vertex set V (G), and let d(u, w) denote the length of a u − w geodesic in G. For k ≥ 1, let dk (x, y) = min{d(x, y), k +1}. A function f: V (G) → Z+ ∪{0} is called a resolving broadcast of G if, for any distinct x, y ∈ V (G), there exists a vertex [Formula Presented]. The broadcast dimension, bdim(G), of G is the minimum of [Formula Presented] over all resolving broadcasts of G, where bcf (G) can be viewed as the total cost of the transmitters (of various strength) used in resolving the entire network described by the graph G. Note that bdim(G) reduces to adim(G) (the adjacency dimension of G, introduced by Jannesari and Omoomi in 2012) if the codomain of resolving broadcasts is restricted to {0, 1}. We determine its value for cycles, paths, and other families of graphs. We prove that bdim(G) = Ω(log n) for all graphs G of order n, and that the result is sharp up to a constant factor. We show that [Formula Presented] and can both be arbitrarily large, where dim(G) denotes the metric dimension of G. We also examine the effect of vertex deletion on both the adjacency dimension and the broadcast dimension of graphs

Three Edge-disjoint Plane Spanning Paths in a Point Set

We study the following problem: Given a set $S$ of $n$ points in the plane, how many edge-disjoint plane straight-line spanning paths of $S$ can one draw? A well known result is that when the $n$ points are in convex position, $\lfloor n/2\rfloor$ such paths always exist, but when the points of $S$ are in general position the only known construction gives rise to two edge-disjoint plane straight-line spanning paths. In this paper, we show that for any set $S$ of at least ten points, no three of which are collinear, one can draw at least three edge-disjoint plane straight-line spanning paths of~$S$. Our proof is based on a structural theorem on halving lines of point configurations and a strengthening of the theorem about two spanning paths, which we find interesting in its own right: if $S$ has at least six points, and we prescribe any two points on the boundary of its convex hull, then the set contains two edge-disjoint plane spanning paths starting at the prescribed points.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Combinatorics

This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics

Perfect Roman Domination and Unique Response Roman Domination

The idea of enumeration algorithms with polynomial delay is to polynomially bound the running time between any two subsequent solutions output by the enumeration algorithm. While it is open for more than four decades if all minimal dominating sets of a graph can be enumerated in output-polynomial time, it has recently been proven that pointwise-minimal Roman dominating functions can be enumerated even with polynomial delay. The idea of the enumeration algorithm was to use polynomial-time solvable extension problems. We use this as a motivation to prove that also two variants of Roman dominating functions studied in the literature, named perfect and unique response, can be enumerated with polynomial delay. This is interesting since Extension Perfect Roman Domination is W[1]-complete if parameterized by the weight of the given function and even W[2]-complete if parameterized by the number vertices assigned 0 in the pre-solution, as we prove. Otherwise, efficient solvability of extension problems and enumerability with polynomial delay tend to go hand-in-hand. We achieve our enumeration result by constructing a bijection to Roman dominating functions, where the corresponding extension problem is polynomimaltime solvable. Furthermore, we show that Unique Response Roman Domination is solvable in polynomial time on split graphs, while Perfect Roman Domination is NP-complete on this graph class, which proves that both variations, albeit coming with a very similar definition, do differ in some complexity aspects. This way, we also solve an open problem from the literature

Combinatorics, Probability and Computing

One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly effective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry