Related papers: SSD Set System, Graph Decomposition and Hamiltonia…

A subset of vertices $F$ in a graph $G$ is called a \emph{dissociation set} if the induced subgraph $G[F]$ of $G$ has maximum degree at most 1. A \emph{maximal dissociation set} of $G$ is a dissociation set which is not a proper subset of…

A set $X \subseteq V(G)$ in a graph $G$ is $(q,k)$-unbreakable if every separation $(A,B)$ of order at most $k$ in $G$ satisfies $|A \cap X| \leq q$ or $|B \cap X| \leq q$. In this paper, we prove the following result: If a graph $G$…

Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that $\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S| = 1$ for…

A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$…

In this series of papers, the primary goal is to enumerate Hamiltonian cycles (HC's) on the grid cylinder graphs $P_{m+1}\times C_n$, where $n$ is allowed to grow whilst $m$ is fixed. In Part~I, we studied the so-called non-contractible…

We address the problem of computing a Minimal Dominating Set in highly dynamic distributed systems. We assume weak connectivity, i.e., the network may be disconnected at each time instant and topological changes are unpredictable. We make…

We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its $k$-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that…

In this note we prove: {\it Let $D$ be a 2-strong digraph of order $n$ such that its $n-1$ vertices have degrees at least $n+k$ and the remaining vertex $z$ has degree at least $n-k-4$, where $k$ is a positive integer. If $D$ contains a…

A Hamiltonian path in a digraph $D$ in which the initial vertex dominates the terminal vertex is called a Hamiltonian bypass. Let $D$ be a 2-strong digraph of order $p\geq 3$ and let $z$ be some vertex of $D$. Suppose that every vertex of…

Given a graph $G=(V,E)$, a subset $X$ of $V$ is an interval of $G$ provided that for any $a, b\in X$ and $ x\in V \setminus X$, $\{a,x\}\in E$ if and only if $\{b,x\}\in E$. For example, $\emptyset$, $\{x\}(x\in V)$ and $V$ are intervals of…

We consider high-order connectivity in $k$-uniform hypergraphs defined as follows: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We describe the…

Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th…

Finding the diameter of a graph in general cannot be done in truly subquadratic assuming the Strong Exponential Time Hypothesis (SETH), even when the underlying graph is unweighted and sparse. When restricting to concrete classes of graphs…

The canonical tree-decomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the…

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(\delta\geq k+1)}$. Let $m=cn$ and…

We propose a general scheme to construct a Hamiltonian $H_{\text{root}}$ describing a square root of an original Hamiltonian $H_{\text{original}}$ based on the graph theory. The square-root Hamiltonian is defined on the subdivided graph of…

A covering of a digraph $D$ by Hamilton cycles is a collection of directed Hamilton cycles (not necessarily edge-disjoint) that together cover all the edges of $D$. We prove that for $1/2 \geq p\geq \frac{\log^{20} n}{n}$, the random…

Eigenvectors of the Laplacian of a cycle graph exhibit the sinusoidal characteristics of the standard DFT basis, and signals defined on such graphs are amenable to linear shift invariant (LSI) operations. In this paper we propose to reduce…

Given a finite set $V$, a convexity $\mathscr{C}$, is a collection of subsets of $V$ that contains both the empty set and the set $V$ and is closed under intersections. The elements of $\mathscr{C}$ are called convex sets. The digital…

The 2-block intersection graph (2-BIG) of a twofold triple system (TTS) is the graph whose vertex set is composed of the blocks of the TTS and two vertices are joined by an edge if the corresponding blocks intersect in exactly two elements.…