Related papers: On the bar visibility number of complete bipartite…

A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical…

A \textit{$t$-unit-bar representation} of a graph $G$ is an assignment of sets of at most $t$ horizontal unit-length segments in the plane to the vertices of $G$ so that (1) all of the segments are pairwise nonintersecting, and (2) two…

Visibility representation of digraphs was introduced by Axenovich, Beveridge, Hutch\-inson, and West (\emph{SIAM J. Discrete Math.} {\bf 27}(3) (2013) 1429--1449) as a natural generalization of $t$-bar visibility representation of…

While the problem of determining the representation number of an arbitrary word-representable graph is NP-hard, this problem is open even for bipartite graphs. The representation numbers are known for certain bipartite graphs including all…

A bipartite covering of a (multi)graph $G$ is a collection of bipartite graphs, so that each edge of $G$ belongs to at least one of them. The capacity of the covering is the sum of the numbers of vertices of these bipartite graphs. In this…

For a bipartite graph $H$, its linear threshold is the smallest real number $\sigma$ such that every bipartite graph $G = (U \sqcup V, E)$ with unbalanced parts $|V| \gtrsim |U|^\sigma$ and without a copy of $H$ must have a linear number of…

Let $G$ be a simple graph with order $n$ and adjacency matrix $\mathbf{A}(G)$. Let $\phi(G; \lambda)=\det(\lambda I-\mathbf{A}(G))=\sum_{i=0}^n\mathbf{a}_i(G)\lambda^{n-i}$ be the characteristic polynomial of $G$, where $\mathbf{a}_i(G)$ is…

Bipartite graphs model the relationship between two disjoint sets of objects. They have a wide range of applications and are often visualized as a 2-layered drawing, where each set of objects is visualized as a set of vertices (points) on…

{\it A unit cube in $k$-dimension (or a $k$-cube) is defined as the cartesian product $R_1 \times R_2 \times ... \times R_k$, where each $R_i$ is a closed interval on the real line of the form $[a_i, a_i+1]$. The {\it cubicity} of $G$,…

In this paper, we present a complete characterization of mutual-visibility sets in trees. It is shown that a subset $S$ is a mutual-visibility set of a tree $T$ if and only if it coincides with the set of leaves of the Steiner subtree…

A bipartite graph $G = (X \cup Y, E)$ is a 2-layer $k$-planar graph if it admits a drawing on the plane such that the vertices in $X$ and $Y$ are placed on two parallel lines respectively, edges are drawn as straight-line segments, and…

For a connected graph $G$ and $X\subseteq V(G)$, we say that two vertices $u$, $v$ are $X$-visible if there is a shortest $u,v$-path $P$ with $V(P)\cap X \subseteq \{u,v\}$. If every two vertices from $X$ are $X$-visible, then $X$ is a…

In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise…

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number…

We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing…

A bar 1-visibility drawing of a graph $G$ is a drawing of $G$ where each vertex is drawn as a horizontal line segment called a bar, each edge is drawn as a vertical line segment where the vertical line segment representing an edge must…

In this paper, we investigate the problem of finding {\it bisections} (i.e., balanced bipartitions) in graphs. We prove the following two results for {\it all} graphs $G$: (1). $G$ has a bisection where each vertex $v$ has at least $(1/4 -…

A visibility representation is a classical drawing style of planar graphs. It displays the vertices of a graph as horizontal vertex-segments, and each edge is represented by a vertical edge-segment touching the segments of its end vertices;…

In this paper we explore some results concerning the spread of the line and the total graph of a given graph. In particular, it is proved that for an $(n,m)$ connected graph $G$ with $m > n \geq 4$ the spread of $G$ is less than or equal to…

Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $\gamma_{t}$($G$), is the minimum cardinality of a total…