Related papers: Characterising AT-free Graphs with BFS
A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A banner is a graph which consists of a hole on four vertices and a single vertex with precisely one neighbor on the hole. We prove…
We extend our previous algorithm that generates all labeled graphs with a given graphical degree sequence to generate all labeled triangle-free graphs with a given graphical degree sequence. The algorithm uses various pruning techniques to…
We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the…
Nut graphs are graphs whose adjacency matrix is singular with one-dimensional null space spanned by a vector with no zero entries. In a recent paper, Ba\v{s}i\'c, Fowler and Pisanski proved that the automorphism group of a nut graph has…
Given a family $\mathcal{F}$ of graphs, a graph is \emph{$\mathcal{F}$-subgraph-free} if it has no subgraph isomorphic to a member of $\mathcal{F}$. We present a fixed-parameter linear-time algorithm that decides whether a planar graph can…
In this paper, we give a polynomial time algorithm which determines if a given triangle-free graph with no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists.
An ordering of the vertices of a graph is \emph{connected} if every vertex (but the first) has a neighbor among its predecessors. The greedy colouring algorithm of a graph with a connected order consists in taking the vertices in order, and…
A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable $H$-free graphs. Such a characterization was previously known only in the…
The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time $O(n^4)$ whether three given vertices of a graph belong to an induced tree. Here, we study four-in-a-tree for triangle-free graphs. We give a structural answer to the…
The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle-free…
The study of geometric hypergraphs gave rise to the notion of $ABAB$-free hypergraphs. A hypergraph $\mathcal{H}$ is called $ABAB$-free if there is an ordering of its vertices such that there are no hyperedges $A,B$ and vertices…
Characterisations of interval graphs, comparability graphs, co-comparability graphs, permutation graphs, and split graphs in terms of linear orderings of the vertex set are presented. As an application, it is proved that interval graphs,…
A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path…
A simple-triangle graph is the intersection graph of triangles that are defined by a point on a horizontal line and an interval on another horizontal line. The time complexity of the recognition problem for simple-triangle graphs was a…
An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) denote the number of vertices of G. A graph is called claw-free if it does not have a subgraph isomorphic to the graph with 4 vertices…
We study the typical structure and the number of triangle-free graphs with $n$ vertices and $m$ edges where $m$ is large enough so that a typical triangle-free graph has a cut containing nearly all of its edges, but may not be bipartite.…
A \emph{hole} in a graph is an induced cycle with at least 4 vertices. A graph is \emph{even-hole-free} if it does not contain a hole on an even number of vertices. A \emph{pyramid} is a graph made of three chordless paths $P_1 = a \dots…
An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient…
Every triangle-free planar graph on n vertices has an independent set of size at least (n+1)/3, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph G on n vertices and an integer k>=0, decides…
A traversal of a connected graph is a linear ordering of its vertices all of whose initial segments induce connected subgraphs. Traversals, and their refinements such as breadth-first and depth-first traversals, are computed by various…