Related papers: A Note on Graph Burning of Path Forests
The dynamics of infectious diseases spread is crucial in determining their risk and offering ways to contain them. We study sequential vaccination of individuals in networks. In the original (deterministic) version of the Firefighter…
Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in…
Graph clustering is a fundamental computational problem with a number of applications in algorithm design, machine learning, data mining, and analysis of social networks. Over the past decades, researchers have proposed a number of…
Researchers, policy makers, and engineers need to make sense of data on spreading processes as diverse as viral infections, water contamination, and misinformation in social networks. Classical questions include predicting infection…
This paper develops a structural theory of unique shortest paths in real-weighted graphs. Our main goal is to characterize exactly which sets of node sequences, which we call path systems, can be realized as unique shortest paths in a graph…
A longest path in a graph is called a detour. It is easy to see that a connected graph of minimum degree at least $2$ and order at least $4$ has at least $4$ detours. We prove that if the number of detours in such a graph of order at least…
Finding paths in graphs is a fundamental graph-theoretic task. In this work, we we are concerned with finding a path with some constraints on its length and the number of vertices neighboring the path, that is, being outside of and incident…
The Burling sequence is a sequence of triangle-free graphs of increasing chromatic number. Any graph which is an induced subgraph of a graph in this sequence is called a Burling graph. These graphs have attracted some attention because they…
Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the…
The connectivity structure of graphs is typically related to the attributes of the nodes. In social networks for example, the probability of a friendship between two people depends on their attributes, such as their age, address, and…
We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The Path Partition problem (PP) has been studied extensively, as it includes Hamiltonian Path as a special case. The…
Lazy burning is a recently introduced variation of burning where only one set of vertices is chosen to burn in the first round. In hypergraphs, lazy burning spreads when all but one vertex in a hyperedge is burned. The lazy burning number…
We consider a version of the forest fire model on graph $G$, where each vertex of a graph becomes occupied with rate one. A fixed vertex $v_0$ is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied…
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds:…
Graph coloring, also known as vertex coloring, considers the problem of assigning colors to the nodes of a graph such that adjacent nodes do not share the same color. The optimization version of the problem concerns the minimization of the…
Given a graph $G$, we associate a path matrix $P$ whose $(i, j)$ entry represents the maximum number of vertex disjoint paths between the vertices $i$ and $j$, with zeros on the main diagonal. In this note, we resolve four conjectures from…
Combinatorial optimization problems near algorithmic phase transitions represent a fundamental challenge for both classical algorithms and machine learning approaches. Among them, graph coloring stands as a prototypical constraint…
A path graph is the intersection graph of paths in a tree. A directed path graph is the intersection graph of paths in a directed tree. Even if path graphs and directed path graphs are characterized very similarly, their recognition…
We use Hartnell's model for virus spread on a graph, also known as firefighting. For rooted trees, we propose an Unburning Algorithm, a type of greedy algorithm starting from the leaves and working back towards the root. We show that the…
In the last decade it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks: separable elements, with connections (or interactions) between certain pairs of them.…