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Given a graph $G=(V, E)$, the problem of Graph Burning is to find a sequence of nodes from $V$, called a burning sequence, to burn the whole graph. This is a discrete-step process, and at each step, an unburned vertex is selected as an…

The feedback set problems are about removing the minimum number of vertices or edges from a graph to break all its cycles. Much effort has gone into understanding their complexity on planar graphs as well as on graphs of bounded degree. We…

In recent years, the increasing prevalence and intensity of wildfires have posed significant challenges to emergency response teams. The utilization of unmanned aerial vehicles (UAVs), commonly known as drones, has shown promise in aiding…

We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…

Graph isomorphism, subgraph isomorphism, and maximum common subgraphs are classical well-investigated objects. Their (parameterized) complexity and efficiently tractable cases have been studied. In the present paper, for a given set of…

Graph burning is motivated by the spread of social influence, and the burning number measures the speed of the spread. Given that the smallest burning number among the spanning trees of a graph determines the burning number of a connected…

In this paper, we propose a feedback control strategy to protect vulnerable areas from wildfires. We consider a system of coupled partial differential equations (PDEs) that models heat propagation and fuel depletion in wildfires and study…

Temporal graphs (in which edges are active at specified times) are of particular relevance for spreading processes on graphs, e.g.~the spread of disease or dissemination of information. Motivated by real-world applications, modification of…

Chip-firing is a combinatorial game played on an undirected graph in which we place chips on vertices. We study chip-firing on an infinite binary tree in which we add a self-loop to the root to ensure each vertex has degree 3. A vertex can…

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set…

The lack of extensive research in the application of inexpensive wireless sensor nodes for the early detection of wildfires motivated us to investigate the cost of such a network. As a first step, in this paper we present several results…

Wildfires are uncontrolled fires in the environment that can be caused by humans or nature. In 2020 alone, wildfires in California have burned 4.2 million acres, damaged 10,500 buildings or structures, and killed more than 31 people,…

The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) \le \lceil \sqrt{n} \rceil.$ While the conjecture remains open, we prove that it is asymptotically true when the order…

A class of countable infinite graphs with unbounded vertex degree is considered. In these graphs, the vertices of large degree `repel' each other, which means that the path distance between two such vertices cannot be smaller than a certain…

The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt [Dist. Comp. '02] proves that, in an undirected unweighted graph, any replacement shortest path avoiding a failing edge can be expressed as the concatenation of two…

By providing a new framework, we extend previous results on locally checkable problems in bounded treewidth graphs. As a consequence, we show how to solve, in polynomial time for bounded treewidth graphs, double Roman domination and Grundy…

Extremal graph theory studies the maximum or minimum number of subgraphs isomorphic to a prescribed graph under given constraints. \textit{Localization} has recently emerged as a framework that refines such problems by assigning extremal…

The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on $n$ vertices. Under…

In this paper we describe an extension of the Variable Neighbourhood Search (VNS) which integrates the basic VNS with other complementary approaches from machine learning, statistics and experimental algorithmic, in order to produce…

Chernyshev, Rauch, and Rautenbach proved that every connected graph on $n$ vertices with less than $\frac{11}{5}n-\frac{18}{5}$ edges has a vertex cut that induces a forest, and conjectured that the same remains true if the graph has less…