Related papers: Efficient Enumeration of Chordless Cycles
We enumerate chord diagrams without loops and without both loops and parallel chords. We show that the former ones describe Hamiltonian paths in $n$-dimensional octahedrons. The latter ones are also known as shapes. For labelled diagrams we…
A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices,…
Let $G$ be a graph and let $\mathrm{cl}(G)$ be the number of distinct induced cycle lengths in $G$. We show that for $c,t\in \mathbb N$, every graph $G$ that does not contain an induced subgraph isomorphic to $K_{t+1}$ or $K_{t,t}$ and…
Learning properties of large graphs from samples has been an important problem in statistical network analysis since the early work of Goodman \cite{Goodman1949} and Frank \cite{Frank1978}. We revisit a problem formulated by Frank…
One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two.…
A \emph{long unichord} in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is \emph{long-unichord-free} if it does not contain any long-unichord. We prove a structure theorem for long-unichord-free…
A polynomial time algorithm which detects all paths and cycles of all lengths in form of vertex pairs (start, finish).
We study the \emph{{interval completion}} problem, which asks for the insertion of a set of at most $k$ edges to make a graph of $n$ vertices into an interval graph. We focus on chordal graphs with no small obstructions, where every…
In this paper, a proof of the cycle double cover conjecture is presented. The cycle double cover conjecture purports that if a graph is bridgeless, then there exists a list of cycles in the graph such that every edge in the graph appears in…
Let $G=(V,E)$ be an unweighted undirected graph with $n$ vertices and $m$ edges. Let $g$ be the girth of $G$, that is, the length of a shortest cycle in $G$. We present a randomized algorithm with a running time of $\tilde{O}\big(\ell \cdot…
In order to find Hamiltonian cycle, algorithm should find edges that creates a Hamiltonian cycle. Higher number of edges creates more possibilities to check to solve the problem. Algorithm rests on analysis of original graph and opposite…
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free…
The Cycle double cover (CDC) conjecture states that for every bridgeless graph $G$, there exists a family $\mathcal{F}$ of cycles such that each edge of the graph is contained in exactly two members of $\mathcal{F}$. Given an embedding of a…
A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node…
In this paper, a polynomial time algorithm for finding the set of all cyclic subsets in a graph is presented. The concept of cyclic subsets has already been introduced in an earlier paper. The algorithm finds cyclic subsets in a graph G by…
We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second…
Circle graphs are the intersection graphs of chords in a circle. This paper presents the first sub-quadratic recognition algorithm for the class of circle graphs. Our algorithm is O(n + m) times the inverse Ackermann function, {\alpha}(n +…
A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan,…
We compute the number of equivalence classes of nonperiodic covering cycles of given length in a non oriented connected graph. A covering cycle is a closed path that traverses each edge of the graph at least once. A special case is the…
We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.