Related papers: Contact Representations of Sparse Planar Graphs
A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An…
A graph is called a strong (resp. weak) bar 1-visibility graph if its vertices can be represented as horizontal segments (bars) in the plane so that its edges are all (resp. a subset of) the pairs of vertices whose bars have a…
We consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry…
Given a planar graph $G$, we consider drawings of $G$ in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding $\pi$ of the vertex set of $G$ into…
In the area of beyond-planar graphs, i.e. graphs that can be drawn with some local restrictions on the edge crossings, the recognition problem is prominent next to the density question for the different graph classes. For 1-planar graphs,…
A $k$-subcoloring of a graph is a partition of the vertex set into at most $k$ cluster graphs, that is, graphs with no induced $P_3$. 2-subcoloring is known to be NP-complete for comparability graphs and three subclasses of planar graphs,…
A Graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their complements. Then we split characterizing…
The Local Computation Algorithms (LCA) model is a computational model aimed at problem instances with huge inputs and output. For graph problems, the input graph is accessed using probes: strong probes (SP) specify a vertex $v$ and receive…
Dimensionality reduction, cluster analysis, and sparse representation are basic components in machine learning. However, their relationships have not yet been fully investigated. In this paper, we find that the spectral graph theory…
We study how well one can recover sparse principal components of a data matrix using a sketch formed from a few of its elements. We show that for a wide class of optimization problems, if the sketch is close (in the spectral norm) to the…
A unit disk graph $G$ on a given set $P$ of points in the plane is a geometric graph where an edge exists between two points $p,q \in P$ if and only if $|pq| \leq 1$. A spanning subgraph $G'$ of $G$ is a $k$-hop spanner if and only if for…
We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold…
We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. For positive we have a construction which shows that all chordal graphs that can be represented as intersection graph of subpaths on a tree…
In this work, we present a theoretical study of signals with sparse representations in the vertex domain of a graph, which is primarily motivated by the discrepancy arising from respectively adopting a synthesis and analysis view of the…
In this paper, motivated by network inference and tomography applications, we study the problem of compressive sensing for sparse signal vectors over graphs. In particular, we are interested in recovering sparse vectors representing the…
We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$,…
The {\sc Plane Diameter Completion} problem asks, given a plane graph $G$ and a positive integer $d$, if it is a spanning subgraph of a plane graph $H$ that has diameter at most $d$. We examine two variants of this problem where the input…
In 1988 Rafla conjectured that every simple drawing of a complete graph $K_n$ contains a plane, i.e., non-crossing, Hamiltonian cycle. The conjecture is far from being resolved. The lower bounds for plane paths and plane matchings have…
This paper studies optimal-area visibility representations of $n$-vertex outer-1-plane graphs, i.e. graphs with a given embedding where all vertices are on the boundary of the outer face and each edge is crossed at most once. We show that…
In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index…