Related papers: Finding Fullerene Patches in Polynomial Time

In this paper, we show that given a weighted, directed planar graph $G$, and any $\epsilon >0$, there exists a polynomial time and $O(n^{\frac{1}{2}+\epsilon})$ space algorithm that computes the shortest path between two fixed vertices in…

The C-Planarity problem asks for a drawing of a $\textit{clustered graph}$, i.e., a graph whose vertices belong to properly nested clusters, in which each cluster is represented by a simple closed region with no edge-edge crossings, no…

In a recent paper, Francis, Illickan, Jose and Rajendraprasad showed that every $n$-vertex plane graph $G$ has (under some natural restrictions) a vertex-partition into two sets $V_1$ and $V_2$ such that each $V_i$ is \emph{dominating}…

The \emph{total graph} $T(G)$ of a multigraph $G$ has as its vertices the set of edges and vertices of $G$ and has an edge between two vertices if their corresponding elements are either adjacent or incident in $G$. We show that if $G$ has…

We consider problems of finding a maximum size/weight $t$-matching without forbidden subgraphs in an undirected graph $G$ with the maximum degree bounded by $t+1$, where $t$ is an integer greater than $2$. Depending on the variant forbidden…

A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to…

Let $P$ be an arbitrary finite partially ordered set. It will be proved that the number of edges of the order polytope ${\mathcal O}(P)$ is equal to that of the chain polytope ${\mathcal C}(P)$. Furthermore, it will be shown that the degree…

For a graph $G=(V,E)$, its exact-distance square, $G^{[\sharp 2]}$, is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance (exactly) $2$ in $G$. The graph $G$ is an…

This paper investigates an extremely classic NP-complete problem: How to determine if a graph G, where each vertex has a degree of at most 4, can be 3-colorable(The research in this paper focuses on graphs G that satisfy the condition where…

Consider a problem where we are given a bipartite graph H with vertices arranged on two horizontal lines in the plane, such that the two sets of vertices placed on the two lines form a bipartition of H. We additionally require that H admits…

Cadences are structurally maximal arithmetic progressions of indices corresponding to equal characters in an underlying string. This paper provides a polynomial time detection algorithm for 3-cadences in grammar-compressed binary strings.…

In a graph G, a dissociation set is a subset of vertices which induces a subgraph with vertex degree at most 1. Finding a dissociation set of maximum cardinality in a graph is NP-hard even for bipartite graphs and is called the maximum…

Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges…

A fullerene graph is a 3-connected cubic planar graph with pentagonal and hexagonal faces. The leapfrog transformation of a planar graph produces the trucation of the dual of the given graph. A fullerene graph is leapfrog if it can be…

Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the…

A graph $G$ embedded in a surface $S$ is called an $S$-grid when every facial boundary walk has length four, that is, the topological dual graph of $G$ in $S$ is 4-regular. Aside from the case where $S$ is the torus or Klein bottle, an…

The forcing number of a graph with a perfect matching $M$ is the minimum number of edges in $M$ whose endpoints need to be deleted, such that the remaining graph only has a single perfect matching. This number is of great interest in…

Let $G$ be a graph on $n$ vertices. For $i\in \{0,1\}$ and a connected graph $G$, a spanning forest $F$ of $G$ is called an $i$-perfect forest if every tree in $F$ is an induced subgraph of $G$ and exactly $i$ vertices of $F$ have even…

We study the following combinatorial problem. Given a planar graph $G=(V,E)$ and a set of simple cycles $\mathcal C$ in $G$, find a planar embedding $\mathcal E$ of $G$ such that the number of cycles in $\mathcal C$ that bound a face in…

We present a polynomial time algorithm that for any graph G and integer k >= 0, either finds a spanning tree with at least k internal vertices, or outputs a new graph G' on at most 3k vertices and an integer k' such that G has a spanning…