Related papers: On the Relationships between Zero Forcing Numbers …
The independent domination number of a finite graph G is the minimum cardinality of an independent dominating set of vertices. The independent bondage number of G is the minimum cardinality of a set of edges whose deletion results in a…
In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic…
A fair dominating set in a graph $G$ (or FD-set) is a dominating set $S$ such that all vertices not in $S$ are dominated by the same number of vertices from $S$; that is, every two vertices not in $S$ have the same number of neighbors in…
The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph $G$ that does not contain a fixed graph as a minor has crossing number $O(\Delta n)$, where $G$…
In this paper, we showcase the process of using an automated conjecturing program called \emph{TxGraffiti} written and maintained by the second author. We begin by proving a conjecture formulated by \emph{TxGraffiti} that for a claw-free…
Hopping forcing is a single player combinatorial game in which the player is presented a graph on $n$ vertices, some of which are initially blue with the remaining vertices being white. In each round $t$, a blue vertex $v$ with all…
In this paper, the notions of {\em trapping} and {\em confining} the robber on a graph are introduced. We present some structural necessary conditions for graphs $G$ not containing the path on $k$ vertices (referred to as $P_k$-free graphs)…
We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which…
Throttling in graphs optimizes a sum or product of resources used, such as the number of vertices in an initial set, and time required, such as the propagation time, to complete a given task. We introduce a new technique to establish sharp…
In this paper a new parameter for hypergraphs called hypergraph infection is defined. This concept generalizes zero forcing in graphs to hypergraphs. The exact value of the infection number of complete and complete bipartite hypergraphs is…
A track layout of a graph consists of a vertex coloring and a total order of each color class, such that no two edges cross between any two color classes. The track number of a graph is the minimum number of colors required by a track…
The forcing number of a perfect matching $M$ of a graph $G$ is the smallest cardinality of subsets of $M$ that are contained in no other perfect matchings of $G$. The forcing spectrum of $G$ is the collection of forcing numbers of all…
We prove that every connected triangle-free graph on $n$ vertices contains an induced tree on $\exp(c\sqrt{\log n})$ vertices, where $c$ is a positive constant. The best known upper bound is $(2+o(1))\sqrt n$. This partially answers…
Let ${\mathcal G}$ be an infinite family of connected graphs and let $k$ be a positive integer. We say that $k$ is ${\it forcing}$ for ${\mathcal G}$ if for all $G \in {\mathcal G}$ but finitely many, the following holds. Any…
A dissociation set of a graph is a set of vertices which induces a subgraph with maximum degree less than or equal to one. The dissociation number of a graph is the maximum cardinality of its dissociation sets. In this paper, we study the…
For any graph $G$ of order $p$, a bijection $f: V(G)\to [1,p]$ is called a numbering of the graph $G$ of order $p$. The strength $str_f(G)$ of a numbering $f: V(G)\to [1,p]$ of $G$ is defined by $str_f(G) = \max\{f(u)+f(v)\; |\; uv\in…
The slope number of a graph $G$ is the smallest number of slopes needed for the segments representing the edges in any straight-line drawing of $G$. It serves as a measure of the visual complexity of a graph drawing. Several bounds on the…
A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a…
A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges. The $t$-tessellability problem aims to decide whether there is a…
Harary et al. and Klein and Randic proposed the forcing number of a perfect matching in mathematics and chemistry, respectively. In detail, the forcing number of a perfect matching M of a graph G is the smallest cardinality of subsets of M…