相关论文: A structural duality for path-decompositions into …

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathscr{I}(G)$, is the graph whose vertices correspond to the…

A \emph{self-complementary} graph is a graph isomorphic to its complement. An isomorphism between $G$ and its complement, viewed as a permutation of $V(G)$, is then called an \emph{antimorphism}. A \emph{skew partition} of $G$ is a…

A graph $G$ is said to be determined by the spectrum of its Laplacian matrix (DLS) if every graph with the same spectrum is isomorphic to $G$. van Dam and Haemers (2003) conjectured that almost all graphs have this property, but that is…

Aboulker, Adler, Kim, Sintiari, and Trotignon conjectured that every graph with bounded maximum degree and large treewidth must contain, as an induced subgraph, a large subdivided wall, or the line graph of a large subdivided wall. This…

Let $G$ be a graph with a vertex set $V$. The graph $G$ is path-proximinal if there are a semimetric $d \colon V \times V \to [0, \infty[$ and disjoint proximinal subsets of the semimetric space $(V, d)$ such that $V = A \cup B$, and…

We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…

Let $G = V, E$ be a simple connected undirected graph. A set $X \subseteq V$ is \emph{geodesically convex} if for any pair of vertices $x, y \in X$, all vertices on all shortest paths in $G$ from $x$ to $y$ are contained in $X$. A set $H…

For a $2$-connected graph $G$ and vertices $u,v$ of $G$ we define an abstract graph $\mathcal{P}(G_{uv})$ whose vertices are the paths joining $u$ and $v$ in $G$, where paths $S$ and $T$ are adjacent if $T$ is obtained from $S$ by replacing…

In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization of several results by various researchers. Let $G$ be a multigraph in which no quadrilaterals share edges with…

A path separator of a graph $G$ is a set of paths $\mathcal{P}=\{P_1,\ldots,P_t\}$ such that for every pair of edges $e,f\in E(G)$, there exist paths $P_e,P_f\in\mathcal{P}$ such that $e\in E(P_e)$, $f\not\in E(P_e)$, $e\not\in E(P_f)$ and…

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar…

The $3$-path isolation number of a connected $n$-vertex graph $G$, denoted by $\iota(G,P_3)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$ intersects the vertex sets…

We show that $k$-uniform hypergraphs on $n$ vertices whose codegree is at least $(2/3 + o(1))n$ can be decomposed into tight cycles, subject to the trivial divisibility conditions. As a corollary, we show those graphs contain tight Euler…

In this paper we investigate results of the form "every graph $G$ has a cycle $C$ such that the induced subgraph of $G$ on $V(G)\setminus V(C)$ has small maximum degree." Such results haven't been studied before, but are motivated by the…

We prove that any graph excluding $K_r$ as a minor has can be partitioned into clusters of diameter at most $\Delta$ while removing at most $O(r/\Delta)$ fraction of the edges. This improves over the results of Fakcharoenphol and Talwar,…

A $P_\ell$-decomposition of a graph $G$ is a set of paths with $\ell$ edges in $G$ that cover the edge set of $G$. Favaron, Genest, and Kouider (2010) conjectured that every $(2k+1)$-regular graph that contains a perfect matching admits a…

A graph is locally chordal if each of its small-radius balls is chordal. In an earlier work [AKK25], the authors and Kobler proved that locally chordal graphs can be characterized by having chordal local covers, by forbidding short cycles…

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…

As a major step in their proof of Wagner's conjecture, Robertson and Seymour showed that every graph not containing a fixed graph $H$ as a minor has a tree-decomposition in which each torso is almost embeddable in a surface of bounded…

A strongly separating path system in a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that, for every two edges $e$ and $f$ of $G$, there is a paths in $\mathcal{P}$ with $e$ and not $f$, and vice-versa. The minimum number of…