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We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120-degree angles between any two edge segments meeting at a vertex or a bend. We show that every graph of maximum…
We discuss the recently introduced concept of k-in-out graphs, and provide a construction for k-in-out graphs for any positive integer k. We derive a lower bound for the number of vertices of a k-in-out graph for any positive integer k, and…
Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any…
An induced matching $M$ in a graph $G$ is a matching in $G$ that is also the edge set of an induced subgraph of $G$. That is, any edge not in $M$ must have no more than one incident vertex saturated by $M$. The maximum size $|M|$ of an…
A graph is maximal $k$-degenerate if each induced subgraph has a vertex of degree at most $k$ and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal…
We examine the Maximum Independent Set Problem in an undirected graph. The main result is that this problem can be considered as the solving the same problem in a subclass of the weighted normal twin-orthogonal graphs. The problem is…
An $(r,z,k)$-mixed graph $G$ has every vertex with undirected degree $r$, directed in- and out-degree $z$, and diameter $k$. In this paper, we study the case $r=z=1$, proposing some new constructions of $(1,1,k)$-mixed graphs with a large…
In this paper, we obtain two spectral upper bounds for the $k$-independence number of a graph which is is the maximum size of a set of vertices at pairwise distance greater than $k$. We construct graphs that attain equality for our first…
Motivated by an application in network security, we investigate the following "linear" case of Directed Mutlicut. Let $G$ be a directed graph which includes some distinguished vertices $t_1, \ldots, t_k$. What is the size of the smallest…
We consider graphs with vertices of degree 1 or 2 and prove that the numbers of components of sizes 2 to q have a limit normal distribution for any q > 1. The result is also extended to multigraphs.
A graph {\it has cutwidth at most 2} if one can number its vertices by $1,\ldots n$ so that for every $i=1,\ldots,n-1$ there are at most 2 edges $(u,v)$ such that $u\le i<v$. A characterization of graphs having cutwidth at most 2 in terms…
Consider the family of all finite graphs with maximum degree $\Delta(G)<d$ and matching number $\nu(G)<m$. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all…
In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard…
Strict outerconfluent drawing is a style of graph drawing in which vertices are drawn on the boundary of a disk, adjacencies are indicated by the existence of smooth curves through a system of tracks within the disk, and no two adjacent…
For a positive integer $k\ge 1$, a graph $G$ is $k$-stepwise irregular ($k$-SI graph) if the degrees of every pair of adjacent vertices differ by exactly $k$. Such graphs are necessarily bipartite. Using graph products it is demonstrated…
This paper considers the degree-diameter problem for undirected circulant graphs. The focus is on extremal graphs of given (small) degree and arbitrary diameter. The published literature only covers graphs of up to degree 7. The approach…
In a directed graph, the imbalance of a vertex is its outdegree minus its indegree. We characterize the sequences that are realizable as the sequence of imbalances of a simple directed graph. Moreover, a realization of a realizable sequence…
Consider a graph $G$ and a real-valued function $f$ defined on the degree set of $G$. The sum of the outputs $f(d_v)$ over all vertices $v\in V(G)$ of $G$ is usually known as the vertex-degree-function indices and is denoted by $H_f(G)$,…
The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In: Towards a…