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The simple greedy algorithm to find a maximal independent set of a graph can be viewed as a sequential update of a Boolean network, where the update function at each vertex is the conjunction of all the negated variables in its neighbourhood. In general, the convergence of the so-called kernel network is complex. A word (sequence of vertices) fixes the kernel network if applying the updates sequentially according to that word. We prove that determining whether a word fixes the kernel network is coNP-complete. We also consider the so-called permis, which are permutation words that fix the kernel network. We exhibit large classes of graphs that have a permis, but we also construct many graphs without a permis.

Awareness structures by Fagin and Halpern (1988) (FH) feature a syntactic awareness correspondence and accessibility relations modeling implicit knowledge. They are a flexible model of unawareness, and best interpreted from a outside modeler's perspective. Unawareness structures by Heifetz, Meier, and Schipper (2006, 2008) (HMS) model awareness by a lattice of state-spaces and explicit knowledge via a possibility correspondence. They can be interpreted as providing the subjective views of agents. Open questions include (1) how implicit knowledge can be defined in HMS structures, and (2) in which way FH structures can be extended to model the agents' subjective views. In this paper, we address (1) by showing how to derive implicit knowledge from explicit knowledge in HMS models. We also introduce a variant of HMS models that instead of explicit knowledge, takes implicit knowledge and awareness as primitives. Further, we address (2) by introducing a category of FH models that are modally equivalent relative to sublanguages and can be interpreted as agents' subjective views depending on their awareness. These constructions allow us to show an equivalence between HMS and FH models. As a corollary, we obtain soundness and completeness of HMS models with respect to the Logic of Propositional Awareness, based on a language featuring both implicit and explicit knowledge.

A graph $G$ is \emph{locally irregular} if no two of its adjacent vertices have the same degree. In [Fioravantes et al. Complexity of finding maximum locally irregular induced subgraph. {\it SWAT}, 2022], the authors introduced and studied the problem of finding a locally irregular induced subgraph of a given a graph $G$ of maximum order, or, equivalently, computing a subset $S$ of $V(G)$ of minimum order, whose deletion from $G$ results in a locally irregular graph; $S$ is denoted as an \emph{optimal vertex-irregulator of $G$}. In this work we provide an in-depth analysis of the parameterised complexity of computing an optimal vertex-irregulator of a given graph $G$. Moreover, we introduce and study a variation of this problem, where $S$ is a substet of the edges of $G$; in this case, $S$ is denoted as an \emph{optimal edge-irregulator of $G$}. In particular, we prove that computing an optimal vertex-irregulator of a graph $G$ is in FPT when parameterised by the vertex integrity, neighborhood diversity or cluster deletion number of $G$, while it is $W[1]$-hard when parameterised by the feedback vertex set number or the treedepth of $G$. In the case of computing an optimal edge-irregulator of a graph $G$, we prove that this problem is in FPT when parameterised by the vertex integrity of $G$, while it is NP-hard even if $G$ is a planar bipartite graph of maximum degree $4$, and $W[1]$-hard when parameterised by the size of the solution, the feedback vertex set or the treedepth of $G$. Our results paint a comprehensive picture of the tractability of both problems studied here, considering most of the standard graph-structural parameters.

Two graphs are homomorphism indistinguishable over a graph class $\mathcal{F}$, denoted by $G \equiv_{\mathcal{F}} H$, if $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$ for all $F \in \mathcal{F}$ where $\operatorname{hom}(F,G)$ denotes the number of homomorphisms from $F$ to $G$. A classical result of Lov\'{a}sz shows that isomorphism between graphs is equivalent to homomorphism indistinguishability over the class of all graphs. More recently, there has been a series of works giving natural algebraic and/or logical characterizations for homomorphism indistinguishability over certain restricted graph classes. A class of graphs $\mathcal{F}$ is homomorphism-distinguishing closed if, for every $F \notin \mathcal{F}$, there are graphs $G$ and $H$ such that $G \equiv_{\mathcal{F}} H$ and $\operatorname{hom}(F,G) \neq \operatorname{hom}(F,H)$. Roberson conjectured that every class closed under taking minors and disjoint unions is homomorphism-distinguishing closed which implies that every such class defines a distinct equivalence relation between graphs. In this note, we confirm this conjecture for the classes $\mathcal{T}_k$, $k \geq 1$, containing all graphs of tree-width at most $k$. As an application of this result, we also characterize which subgraph counts are detected by the $k$-dimensional Weisfeiler-Leman algorithm. This answers an open question from [Arvind et al., J. Comput. Syst. Sci., 2020].

We consider the problem of fair allocation of $m$ indivisible items to a group of $n$ agents with subsidy (money). Our work mainly focuses on the allocation of chores but most of our results extend to the allocation of goods as well. We consider the case when agents have (general) additive cost functions. Assuming that the maximum cost of an item to an agent can be compensated by one dollar, we show that a total of $n/4$ dollars of subsidy suffices to ensure a proportional allocation. Moreover, we show that $n/4$ is tight in the sense that there exists an instance with $n$ agents for which every proportional allocation requires a total subsidy of at least $n/4$. We also consider the weighted case and show that a total subsidy of $(n-1)/2$ suffices to ensure a weighted proportional allocation.

Given a graph $G=(V,E)$, for a vertex set $S\subseteq V$, let $N(S)$ denote the set of vertices in $V$ that have a neighbor in $S$. Extending the concept of binding number of graphs by Woodall~(1973), for a vertex set $X \subseteq V$, we define the binding number of $X$, denoted by $\bind(X)$, as the maximum number $b$ such that for every $S \subseteq X$ where $N(S)\neq V(G)$ it holds that $|N(S)|\ge b {|S|}$. Given this definition, we prove that if a graph $V(G)$ contains a subset $X$ with $\bind(X)= 1/k$ where $k$ is an integer, then $G$ possesses a matching of size at least $|X|/(k+1)$. Using this statement, we derive tight bounds for the estimators of the matching size in planar graphs. These estimators are previously used in designing sublinear space algorithms for approximating the maching size in the data stream model of computation. In particular, we show that the number of locally superior vertices is a $3$ factor approximation of the matching size in planar graphs. The previous analysis by Jowhari (2023) proved a $3.5$ approximation factor. As another application, we show a simple variant of an estimator by Esfandiari \etal (2015) achieves $3$ factor approximation of the matching size in planar graphs. Namely, let $s$ be the number of edges with both endpoints having degree at most $2$ and let $h$ be the number of vertices with degree at least $3$. We prove that when the graph is planar, the size of matching is at least $(s+h)/3$. This result generalizes a known fact that every planar graph on $n$ vertices with minimum degree $3$ has a matching of size at least $n/3$.

An obstacle representation of a graph $G$ consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of $G$ to points such that two vertices are adjacent in $G$ if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each $n$-vertex graph is $O(n \log n)$ [Balko, Cibulka, and Valtr, 2018] and that there are $n$-vertex graphs whose obstacle number is $\Omega(n/(\log\log n)^2)$ [Dujmovi\'c and Morin, 2015]. We improve this lower bound to $\Omega(n/\log\log n)$ for simple polygons and to $\Omega(n)$ for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of $n$-vertex graphs with bounded obstacle number, solving a conjecture by Dujmovi\'c and Morin. We also show that if the drawing of some $n$-vertex graph is given as part of the input, then for some drawings $\Omega(n^2)$ obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph $G$ is fixed-parameter tractable in the vertex cover number of $G$. Second, we show that, given a graph $G$ and a simple polygon $P$, it is NP-hard to decide whether $G$ admits an obstacle representation using $P$ as the only obstacle.

Several physical problems modeled by second-order partial differential equations can be efficiently solved using mixed finite elements of the Raviart-Thomas family for N-simplexes, introduced in the seventies. In case Neumann conditions are prescribed on a curvilinear boundary, the normal component of the flux variable should preferably not take up values at nodes shifted to the boundary of the approximating polytope in the corresponding normal direction. This is because the method's accuracy downgrades, which was shown in \cite{FBRT}. In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this paper an alternative with straight-edged triangles for two-dimensional problems is proposed. The key point of this method is a Petrov-Galerkin formulation of the mixed problem, in which the test-flux space is a little different from the shape-flux space. After carrying out a well-posedness and stability analysis, error estimates of optimal order are proven.

Graph neural networks generalize conventional neural networks to graph-structured data and have received widespread attention due to their impressive representation ability. In spite of the remarkable achievements, the performance of Euclidean models in graph-related learning is still bounded and limited by the representation ability of Euclidean geometry, especially for datasets with highly non-Euclidean latent anatomy. Recently, hyperbolic space has gained increasing popularity in processing graph data with tree-like structure and power-law distribution, owing to its exponential growth property. In this survey, we comprehensively revisit the technical details of the current hyperbolic graph neural networks, unifying them into a general framework and summarizing the variants of each component. More importantly, we present various HGNN-related applications. Last, we also identify several challenges, which potentially serve as guidelines for further flourishing the achievements of graph learning in hyperbolic spaces.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.