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We intend to create new concepts aimed at finding necessary and sufficient conditions for Boolean satisfiability so that these conditions can be verified in polynomial time. Based on these conditions it will be possible to create an…
Extremal problems involving independent sets are much studied. Two of the most important extremal problems in this context are concerned with the sharp upper bounds for the number of independent sets of fixed size and the independence…
In the Graph Reconstruction (GR) problem, a player initially only knows the vertex set $V$ of an input graph $G=(V, E)$ and is required to learn its set of edges $E$. To this end, the player submits queries to an oracle and must deduce $E$…
We construct several pairwise-incomparable bounds on the projective dimensions of edge ideals. Our bounds use combinatorial properties of the associated graphs; in particular we draw heavily from the topic of dominating sets. Through…
Hypergraphs provide an effective abstraction for modeling multi-way group interactions among nodes, where each hyperedge can connect any number of nodes. Different from most existing studies which leverage statistical dependencies, we study…
The standard axioms of set theory, the Zermelo-Fraenkel axioms (ZFC), do not suffice to answer all questions in mathematics. While this follows abstractly from Kurt G\"odel's famous incompleteness theorems, we nowadays know numerous…
We introduce the concept of the primitivity of independent set in vertex-transitive graphs, and investigate the relationship between the primitivity and the structure of maximum independent sets in direct products of vertex-transitive…
We study recursively defined functions associated with directed graphs on the k dimensional nonnegative integral lattice. The existence of certain combinatorial structures associated with these function classes are shown to be independent…
We investigate when the independence complex of $G[H]$, the lexicographical product of two graphs $G$ and $H$, is either vertex decomposable or shellable. As an application, we construct an infinite family of graphs with the property that…
We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely $f$-Balanced Independent Set ($f$-BIS) and $f$-Balanced Dominating Set ($f$-BDS). Let $G=(V,E)$ be a vertex-colored interval…
The independence complex of a graph is a simplicial complex whose faces correspond to the independent sets of $G$. While independence complexes have been studied extensively for many graph classes, including square grid graphs, relatively…
We study the Dominating set problem and Independent Set Problem for dynamic graphs in the vertex-arrival model. We say that a dynamic algorithm for one of these problems is $k$-stable when it makes at most $k$ changes to its output…
The transversal hypergraph problem is the task of enumerating the minimal hitting sets of a hypergraph. It is a long-standing open question whether this can be done in output-polynomial time. For hypergraphs whose solutions have bounded…
A vertex set $D$ in a finite undirected graph $G$ is an efficient dominating set (e.d.s. for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The Efficient Domination (ED) problem, which asks for the existence…
Incremental Learning scenarios do not always represent real-world inference use-cases, which tend to have less strict task boundaries, and exhibit repetition of common classes and concepts in their continual data stream. To better represent…
A maximal independent set is an independent set that is not a subset of any other independent set. It is also the key problem of mathematics, computer science, and other fields. A counting problem is a type of computational problem that…
In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse…
A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U. A subset W is called a resolving set if every vertex in G is uniquely determined by its…
In this paper we take closer look at recent developments for the chase procedure, and provide additional results. Our analysis allows us create a taxonomy of the chase variations and the properties they satisfy. Two of the most central…
A variant of the well-known Set Covering Problem is studied in this paper, where subsets of a collection have to be selected, and pairwise conflicts among subsets of items exist. The selection of each subset has a cost, and the inclusion of…