Related papers: On Triangle Estimation using Tripartite Independen…
Every triangle-free planar graph on n vertices has an independent set of size at least (n+1)/3, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph G on n vertices and an integer k>=0, decides…
A simple-triangle graph (also known as a PI graph) is the intersection graph of a family of triangles defined by a point on a horizontal line and an interval on another horizontal line. The recognition problem for simple-triangle graphs was…
A set of edges $T$ in a graph $G$ is triangle-independent if $T$ contains at most one edge from each triangle in $G$. Let $\alpha_1(G)$ denote the maximum size of the triangle-independent set in $G$, and let $\tau_B(G)$ denote minimum size…
Let $k\ge 2$ and $n_1\ge n_2\ge n_3\ge n_4$ be integers such that $n_4$ is sufficiently larger than $k$. We determine the maximum number of edges of a 4-partite graph with parts of sizes $n_1,\dots, n_4$ that does not contain $k$…
Understanding the higher-order interactions within network data is a key objective of network science. Surveys of metadata triangles (or patterned 3-cycles in metadata-enriched graphs) are often of interest in this pursuit. In this work, we…
The number of triangles (hereafter denoted by $\Delta$) is an important metric to analyze massive graphs. It is also used to compute clustering coefficient in networks. This paper proposes a new algorithm called PES (Priority Edge Sampling)…
The widely studied edge modification problems ask how to minimally alter a graph to satisfy certain structural properties. In this paper, we introduce and study a new edge modification problem centered around transforming a given graph into…
The clustering coefficient and the transitivity ratio are concepts often used in network analysis, which creates a need for fast practical algorithms for counting triangles in large graphs. Previous research in this area focused on…
Traditional frameworks for dynamic graphs have relied on processing only the stream of edges added into or deleted from an evolving graph, but not any additional related information such as the degrees or neighbor lists of nodes incident to…
The task of listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to…
Graphs and networks are used to model interactions in a variety of contexts. There is a growing need to quickly assess the characteristics of a graph in order to understand its underlying structure. Some of the most useful metrics are…
We use Menger's Theorem and K\"onig's Line Colouring Theorem to show that in any tripartite graph with two complete (bipartite) sides the maximum number of pairwise edge-disjoint triangles equals the minimum number of edges that meet all…
Triangle counting is a fundamental and widely studied problem on static graphs, and recently on temporal graphs, where edges carry information on the timings of the associated events. Streaming processing and resource efficiency are crucial…
Bipartite graphs serve as a natural model for representing relationships between two different types of entities. When analyzing bipartite graphs, butterfly counting is a fundamental research problem that aims to count the number of…
On signed social networks, balanced and unbalanced triangles are a critical motif due to their role as the foundations of Structural Balance Theory. The uses for these motifs have been extensively explored in networks with known edge signs,…
We extend our previous algorithm that generates all labeled graphs with a given graphical degree sequence to generate all labeled triangle-free graphs with a given graphical degree sequence. The algorithm uses various pruning techniques to…
Triangle counting and sampling are two fundamental problems for streaming algorithms. Arguably, designing sampling algorithms is more challenging than their counting variants. It may be noted that triangle counting has received far greater…
We study the problem of approximating the $3$-profile of a large graph. $3$-profiles are generalizations of triangle counts that specify the number of times a small graph appears as an induced subgraph of a large graph. Our algorithm uses…
In this report we present an algorithm solving Triangle Counting in time $O(d^2n+m)$, where n and m, respectively, denote the number of vertices and edges of a graph G and d denotes its twin-width, a recently introduced graph parameter. We…
In general graph theory, the only relationship between vertices are expressed via the edges. When the vertices are embedded in an Euclidean space, the geometric relationships between vertices and edges can be interesting objects of study.…