Urban heat islands pose significant public health concerns, yet neighborhood-scale thermal assessment faces data limitations. Satellite-derived surface temperatures provide spatially complete coverage but measure surface radiative properties rather than pedestrian-level air temperature. Meanwhile, sparse meteorological networks cannot resolve intra-urban heterogeneity. Morphology-based approaches address this gap by using urban form characteristics as physical determinants of local thermal conditions through their influence on ventilation, shading, and heat storage. However, morphological indicators are inherently correlated, causing multicollinearity, and conventional classifications impose rigid boundaries that misrepresent transitional zones. This study develops a framework combining Principal Component Analysis (PCA) and Fuzzy C-Means (FCM) clustering to address both challenges, applied to Warsaw (Poland), using nine urban morphology variables computed at 10 m resolution with a 310 m neighborhood analysis window and strict multicollinearity control (VIF<10). PCA extracted two significant components (eigenvalues>1) explaining 65.6% of variance: PC1 (43.3%) represented the urban-natural gradient, while PC2 (22.3%) distinguished water-influenced zones from vegetated areas. FCM clustering identified nine morphological typologies. A novel membership transformation approach aligned fuzzy membership degrees with relative thermal patterns derived from averaged Landsat surface temperatures (summer 2019-2025), orienting values toward thermal hazard magnitude rather than cluster proximity. The highest-risk cluster (compact mid-/high-rise built-up areas, mean LST=36.0 C-degrees) was further stratified into five thermal hazard classes, revealing high building density (BCR=0.56) and minimal vegetation (0.5%) in Very High hazard zones. The framework provides actionable information for targeted heat mitigation while quantifying classification uncertainty in transitional zones, offering a generalizable approach for cities with adequate morphological data.

The graph decomposition method presented in this paper is centered on the edge-disjoint partitioning of a graph G into subgraphs. The decomposition is called the path decomposition of G when every one of these subgraphs is a path. Of particular significance is the pendant number of G, which is defined as the least number of end vertices of paths inside the given path decomposition. Although Pyber verified Gallai’s conjecture for graphs with an odd-degree vertex in each cycle, his method is ineffective for decomposing such graphs. In this paper, an improved linear-time approach for finding the pendant number of certain particular graph families, called Pyber graphs, is presented.

A finite set of points is generic if no two points are on the same vertical or horizontal line. The set is orthogonally convex if every point has an empty quadrant. We study the smallest integer $N_o(n)$ such that every generic set of $N_o(n)$ points contains a orthogonally convex subset of size $n$. For even $n$, we prove $N_0(n) = \frac{1}{8}(n^2+2n+8)$, which is tight, and in the odd case we get close upper and lower bounds. To prove these results, we apply the theory of Greene and Kleitman (1976) and Frank (1980) about posets to the partial order of point sets in the plane. Generic sets correspond to permutations in a canonical way. A permutation is convex if it is order isomorphic to a finite generic set of points in convex position. The value of $N_o(n)$ is also the smallest $N$ such that every permutation of $N$ contains a convex subpermutation of size $n$.

Let G = (V, E) be a graph and let f: V -> Z(+). An f-matching in G is a set of edges F subset of E such that every vertex v is an element of V is incident to at most f(v) edges. In this paper we will give a constant-time distributed algorithm which approximates a maximum f-matching in bi-colored graphs of constant arboricity.

The increasing number of extreme weather events (EWEs) poses a challenge for communities and agencies responsible for risk management. While large cities receive attention due to their significant exposure, there is less research concerning resilience on the local level, where losses can be smaller compared with large cities but severe for small communities. This study explores how resilience built by eight small Polish municipalities endangered by extreme weather events enables them to cope with a significant extreme event - the 2017 storm. The qualitative comparative analysis (QCA) method was applied, and factor conditions impacting the municipalities' resilience were identified. The analysis revealed that lack of the municipality's administrative centrality appeared to be a sufficient factor for high local resilience, as well as a combination of factors: the presence of institutional memory, the municipality's precautionary approach, non-resistance arrangement type, and the developed social coping capacity. The implications of the study are discussed.

Cop Robber game is a two player game played on an undirected graph. In this game, the cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. We study cop-edge critical graphs, i.e. graphs G such that for any edge e in E(G) either c(G - e) < c(G) or c(G > e) > c(G). In this article, we study the edge criticality of generalized Petersen graphs and Paley graphs.

We prove that a simple distributed algorithm finds a constant approximation of an optimal distance- k dominating set in graphs with no K 2 , t-minor. The algorithm runs in a constant number of rounds. We further show how this procedure can be used to give a distributed algorithm which given ϵ > 0 and k , t ∈ Z + finds in a graph G = ( V , E ) with no K 2 , t-minor a distance- k dominating set of size at most ( 1 + ϵ ) of the optimum. The algorithm runs in O ( log ⁎ ⁡ | V | ) rounds in the Local model. In particular, both algorithms work in outerplanar graphs.

We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.

Felsner, Li and Trotter showed that the dimension of the adjacency poset of an outerplanar graph is at most 5, and gave an example of an outerplanar graph whose adjacency poset has dimension 4. We improve their upper bound to 4, which is then best possible.

We give a distributed algorithm which given ϵ > 0 finds a (1 − ϵ )-factor approximation of a maximum f -matching in graphs G = ( V, E ) of sub-logarithmic expansion. Using a similar approach we also give a distributed approximation of a maximum b -matching in the same class of graphs provided the function b : V → Z + is L -Lipschitz for some constant L . Both algorithms run in O (log ∗ n ) rounds in the LOCAL model, which is optimal.

A k-dominating set in a graph G=(V,E) is a set U⊆V such that every vertex of G is either in U or has at least k neighbors in U. In this paper we give simple distributed approximation algorithms in the standard Local model of computations for the minimum k-dominating set problem for k≥2 in graphs with no K3,h-minor for some h∈Z+ and graphs with no K4,4-minor. In particular, this gives fast distributed approximations for graphs of bounded genus and linklessly embeddable graphs. The algorithms give a constant approximation ratio and run in a constant number of rounds. In addition, we will give a (1+ϵ)-approximation for an arbitrary fixed ϵ>0 which runs in O(log⁎⁡n) rounds where n is the order of a graph.

We give a distributed algorithm which finds a constant approximation of a minimum dominating set in graphs of constant genus. The algorithm works in the CONGESTBC model (distributed, synchronous, with short messages and broadcast type of transmissions) and runs in a constant number of distributed rounds.

Cops and Robbers is a two player game played on an undirected graph. In this game the cops try to capture a robber moving on the vertices of a graph. The cop number of a graph, denoted by c(G), is the least number of cops needed to guarantee that the robber will be caught. In this paper we present results concerning games on G(Xi), that is the graph obtained by connecting the corresponding vertices in G and its complement (G) over bar. In particular we show that for planar graphs c(G(Xi)) <= 3. Furthermore we investigate the cop edge-critical graphs, i.e. graphs that for any edge e in G we have either c(G - e) < c(G) or c(G - e) > c(G). We show a couple of examples of cop edge-critical graphs having cop number equal to 3.

In this paper we consider the 2-dominating set problem (2MDS). We look for a smallest subset of vertices D⊆V with the property that every vertex in V∖D is adjacent to at least 2 vertices of D. We are interested in the distributed complexity of this problem in the local model, where the nodes have no identifiers but there is a port ordering available. We propose a distributed local (constant time) algorithm yielding a 6-approximation in the class of planar graphs. Earlier result shows that in this case, for any ϵ>0, there is no deterministic distributed local/constant-round algorithm providing a (5−ϵ)-approximation of the 2MDS.

For a graph G the random n-lift of G is obtained by replacing each of its vertices by a set of n vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of G are adjacent. We show that asymptotically almost surely the random lift of a graph G is Hamiltonian, provided G has the minimum degree at least 5 and contains two disjoint Hamiltonian cycles whose union is not a bipartite graph.

In this paper we consider a generalization of the classical dominating set problem to the k-tuple dominating set problem (kMDS). For any positive integer k, we look for a smallest subset of vertices D ⊆ V with the property that every vertex in V ∖ D is adjacent to at least k vertices of D. We are interested in the distributed complexity of this problem in the model, where the nodes have no identifiers. The most challenging case is when k = 2, and for this case we propose a distributed local algorithm, which runs in a constant number of rounds, yielding a 7-approximation in the class of planar graphs. On the other hand, in the class of algorithms in which every vertex uses only its degree and the degree of its neighbors to make decisions, there is no algorithm providing a (5 − ε)-approximation of the 2MDS problem. In addition, we show a lower bound of (4 − ε) for the 2MDS problem even if unique identifiers are allowed. For k ≥ 3, we show that for the problem kMDS in planar graphs, a trivial algorithm yields a k/(k − 2)-approximation. In the model with unique identifiers this, surprisingly, is optimal for k = 3,4,5, and 6, as we provide a matching lower bound.

In this note we study asymptotic properties of random lifts of graphs introduced by Amit and Linial as a new model of random graphs. Given a base graph $G$ and an integer $n$, a random lift of $G$ is obtained by replacing each vertex of $G$ by a set of $n$ vertices, and joining these sets by random matchings whenever the corresponding vertices of $G$ are adjacent. In this paper we study connectivity properties of random lifts. We show that the size of the largest topological clique in typical random lifts, with $G$ fixed and $n\rightarrow\infty$, is equal to the maximum degree of the core of $G$ plus one. A similar idea can be used to prove that for any graph $G$ with $\delta(G)\geq2k-1$ almost every random lift of $G$ is $k$-linked.

We consider approximate strong equilibria (SE) in strategic job scheduling games with two uniformly related machines. Jobs are assigned to machines, and each job wishes to minimize its cost, given by the completion time of the machine it is assigned to. Finding a Nash equilibrium (NE) in this game is simple. However, NE-configurations are not stable against coordinated deviations of several jobs. Various measures can be used to evaluate how well an NE-configuration approximates SE.A schedule is said to be an alpha-SE if there is no coalitional deviation such that every member of the coalition reduces its cost by a factor greater than alpha. We show that any pure NE on two related machines of speed ratio s is a s(2)+s-1/s(2) <= 5/4-SE, and provide a matching lower bound. In addition, we show that the LPT (Longest Processing Time) algorithm provides a better approximation ratio than a general NE, in particular, any LPT schedule is a 1.1011-SE. This is in contrast to the LS (List Scheduling) greedy algorithm for which the improvement ratio of coalitional deviations can be arbitrarily large. In addition, we design a fully polynomial time approximation scheme (FPTAS), which computes an NE that is a (1 + epsilon)-SE.We also provide bounds for two other measures of approximate SE, considering the supremum possible improvement of a deviating job and the maximal increase in the cost of non-coalition members, and show that checking whether a specific schedule is an SE is co-NP-complete, which motivates the study of approximate strong equilibria.Finally, we consider multiple machines. We show that any pure NE on m related machines is a 2-SE. We give improved results for identical machines, and in particular, we show that any pure NE is a 1.32-SE. (c) 2013 Elsevier B.V. All rights reserved.

In the thesis we study selected properties of random coverings of graphs introduced by Amit and Linial in 2002. A random n-covering of a graph G, denoted by G, is obtained by replacing each vertex v of G by an n-element set Gv and then choosing, independently for every edge e = {x, y} ∈ E(G), uniformly at random a perfect matching between Gx and Gy. The first problem we consider is the typical size of the largest topological clique in a random covering of given graph G. We show that asymptotically almost surely a random n-covering G of a graph G contains the largest topological clique which is allowed by the structure of G. The second property we examine is the existence of a Hamilton cycle in G. We show that if G has minimum degree at least 5 and contains two edge disjoint Hamilton cycles whose union is not a bipartite graph, then asymptotically almost surely G is Hamiltonian.