Constructing dense graphs with sublinear Hadwiger number

Mader asked to explicitly construct dense graphs for which the size of the largest clique minor is sublinear in the number of vertices. Such graphs exist as a random graph almost surely has this property. This question and variants were popularized by Thomason over several articles. We answer these questions by showing how to explicitly construct such graphs using blow-ups of small graphs with this property. This leads to the study of a fractional variant of the clique minor number, which may be of independent interest.Comment: 10 page

Twin subgraphs and core-semiperiphery-periphery structures

A standard approach to reduce the complexity of very large networks is to group together sets of nodes into clusters according to some criterion which reflects certain structural properties of the network. Beyond the well-known modularity measures defining communities, there are criteria based on the existence of similar or identical connection patterns of a node or sets of nodes to the remainder of the network. A key notion in this context is that of structurally equivalent or twin nodes, displaying exactly the same connection pattern to the remainder of the network. The first goal of this paper is to extend this idea to subgraphs of arbitrary order of a given network, by means of the notions of T-twin and F-twin subgraphs. This is motivated by the need to provide a systematic approach to the analysis of core-semiperiphery-periphery (CSP) structures, a notion which somehow lacks a formal treatment in the literature. The goal is to provide an analytical framework accommodating and extending the idea that the unique (ideal) core-periphery (CP) structure is a 2-partitioned K2. We provide a formal definition of CSP structures in terms of core eccentricities and periphery degrees, with semiperiphery vertices acting as intermediaries. The T-twin and F-twin notions then make it possible to reduce the large number of resulting structures by identifying isomorphic substructures which share the connection pattern to the remainder of the graph, paving the way for the decomposition and enumeration of CSP structures. We compute the resulting CSP structures up to order six. We illustrate the scope of our results by analyzing a subnetwork of the network of 1994 metal manufactures trade. Our approach can be further applied in complex network theory and seems to have many potential extensions

Consistent random vertex-orderings of graphs

Given a hereditary graph property $\mathcal{P}$, consider distributions of random orderings of vertices of graphs $G\in\mathcal{P}$ that are preserved under isomorphisms and under taking induced subgraphs. We show that for many properties $\mathcal{P}$ the only such random orderings are uniform, and give some examples of non-uniform orderings when they exist

Hitting and Harvesting Pumpkins

The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges. A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on covering and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time 2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be covered by at most k vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c=1,2 respectively. On the other hand, we present a O(log n)-approximation algorithm for both the problems of covering all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models.Comment: v2: several minor change

Finding topological subgraphs is fixed-parameter tractable

We show that for every fixed undirected graph $H$, there is a $O(|V(G)|^3)$ time algorithm that tests, given a graph $G$, if $G$ contains $H$ as a topological subgraph (that is, a subdivision of $H$ is subgraph of $G$). This shows that topological subgraph testing is fixed-parameter tractable, resolving a longstanding open question of Downey and Fellows from 1992. As a corollary, for every $H$ we obtain an $O(|V(G)|^3)$ time algorithm that tests if there is an immersion of $H$ into a given graph $G$. This answers another open question raised by Downey and Fellows in 1992

Large components in random induced subgraphs of n-cubes

In this paper we study random induced subgraphs of the binary $n$-cube, $Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with independent probability $\lambda_n$. Using a novel construction of subcomponents we study the largest component for $\lambda_n=\frac{1+\chi_n}{n}$, where $\epsilon\ge \chi_n\ge n^{-{1/3}+ \delta}$, $\delta>0$. We prove that there exists a.s. a unique largest component $C_n^{(1)}$. We furthermore show that $\chi_n=\epsilon$, $| C_n^{(1)}|\sim \alpha(\epsilon) \frac{1+\chi_n}{n} 2^n$ and for $o(1)=\chi_n\ge n^{-{1/3}+\delta}$, $| C_n^{(1)}| \sim 2 \chi_n \frac{1+\chi_n}{n} 2^n$ holds. This improves the result of \cite{Bollobas:91} where constant $\chi_n=\chi$ is considered. In particular, in case of $\lambda_n=\frac{1+\epsilon} {n}$, our analysis implies that a.s. a unique giant component exists.Comment: 18 Page