Related papers: The Path-Extremal Conjecture for Zero Forcing: Dis…

A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most (n+1)/2.…

An elimination tree of a connected graph $G$ is a rooted tree on the vertices of $G$ obtained by choosing a root $v$ and recursing on the connected components of $G-v$ to obtain the subtrees of $v$. The graph associahedron of $G$ is a…

Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the \textsc{Central path} (CP) problem our aim is to find a path of minimum eccentricity. This…

Amos et al. (Discrete Appl. Math. 181 (2015) 1-10) introduced the notion of the $k$-forcing number of graph for a positive integer $k$ as the generalization of the zero forcing number of a graph. The $k$-forcing number of a simple graph…

An efficient implicit representation of an $n$-vertex graph $G$ in a family $\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency between every pair of vertices can be determined…

The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph,…

In 2006, Bar\'at and Thomassen posed the following conjecture: for each tree $T$, there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(G)|$ is divisible by $|E(T)|$, then $G$ admits a decomposition…

In any graph, the maximum size of an induced path is bounded by the maximum size of a path. However, in the general case, one cannot find a converse bound, even up to an arbitrary function, as evidenced by the case of cliques. Galvin, Rival…

The central path problem is a variation on the single facility location problem. The aim is to find, in a given connected graph $G$, a path $P$ minimizing its eccentricity, which is the maximal distance from $P$ to any vertex of the graph…

A graph is one-ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex $v$ {\em dominates} a ray in the…

We provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. In general, this degree…

A graph class is monotone if it is closed under taking subgraphs. It is known that a monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a so-called tripod, that is, a…

A path (resp. cycle) decomposition of a graph $G$ is a set of edge-disjoint paths (resp. cycles) of $G$ that covers the edge set of $G$. Gallai (1966) conjectured that every graph on $n$ vertices admits a path decomposition of size at most…

Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded…

We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main result…

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…

We solve a conjecture by Becker et al. (arXiv:2404.05963) on the topic of zero forcing regarding the number of minimal forts of a tree. They conjectured and we prove $\mathcal{F}_{T_n} \le \binom{n}{2} \mathcal{F}_{P_n}$ where…

The canonical tree-decomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the…

For a graph $G$, $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ denotes the size of the largest clique in $G$. A hereditary class of graphs is called $\chi$-bounded if there is a function $f$ such that for each graph $G$ in…

Gallai's conjecture asserts that every connected graph on $n$ vertices can be decomposed into $\frac{n+1}{2}$ paths. For general graphs (possibly disconnected), it was proved that every graph on $n$ vertices can be decomposed into…