Related papers: Cycle decompositions: from graphs to continua
End-spaces of infinite graphs naturally generalise the Freudenthal boundary and sit at the interface between graph theory, geometric group theory and topology. Our main result is that every end-space can topologically be represented by a…
A new general decomposition theory inspired from modular graph decomposition is presented. This helps unifying modular decomposition on different structures, including (but not restricted to) graphs. Moreover, even in the case of graphs,…
Let $G$ be a simple and finite graph. A graph is said to be \textit{decomposed} into subgraphs $H_1$ and $H_2$ which is denoted by $G= H_1 \oplus H_2$, if $G$ is the edge disjoint union of $H_1$ and $H_2$. If $G= H_1 \oplus H_2 \oplus H_3…
In this paper, the concept of cyclic subsets in graph theory is introduced. An interesting theorem which relates to the collective Hamiltonicity of these cyclic subsets in graphs is also presented. This paper uses this theorem to construct…
It is folklore that the cycle space of graphs of polytopes is generated by the cycles bounding the 2-faces. We provide a proof of this result that bypass homological arguments, which seem to be the most widely known proof. As a corollary,…
A cyclic subgroup graph of a group $G$ is a graph whose vertices are cyclic subgroups of $G$ and two distinct vertices $H_1$ and $H_2$ are adjacent if $H_1\leq H_2$, and there is no subgroup $K$ such that $H_1<K<H_2$. M.T\u{a}rn\u{a}uceanu…
Many systems comprising entities in interactions can be represented as graphs, whose structure gives significant insights about how these systems work. Network theory has undergone further developments, in particular in relation to…
For simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex whose vertices are the graph homomorphisms $G\to H$ and whose edges connect the pairs of homomorphisms which differ in a single vertex of $G$. Hom…
The deck of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}] \colon x \in X\}$, where $[Z]$ denotes the homeomorphism class of $Z$. A space $X$ is topologically reconstructible if whenever…
A graph \textit{G} is a tuple (\textit{V}, \textit{E}), where \textit{V} is the vertex set, \textit{E} is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree 2 on an…
We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.
In a balanced graph decomposition, every vertex of the host graph appears in the same number of blocks. We propose the use of colored loops as a framework for unifying various other types of local balance conditions in graph decompositions.…
In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set of a graph into paths and cycles of a given minimum length. Our main contribution is the…
Network centrality measures play a crucial role in understanding graph structures, assessing the importance of nodes, paths, or cycles based on directed or reciprocal interactions encoded by vertices and edges. Estrada and Ross extended…
In this paper, a new general decomposition theory inspired from modular graph decomposition is presented. Our main result shows that, within this general theory, most of the nice algorithmic tools developed for modular decomposition are…
We generalise structure tree theory, which is based on removing finitely many edges, to removing finitely many vertices. This gives a significant generalization of Tutte's tree decomposition of 2-connected graphs into 3-connected blocks.…
If the line graph of a graph $G$ decomposes into Hamiltonian cycles, what is $G$? We answer this question for decomposition into two cycles.
Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we…
We associate a graph $\Gamma_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | \left<x,y\right> \text{is cyclic for all} y\in G\}$, and…
Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features,…