Related papers: Piercing independent sets in graphs without large …

A matching $M$ in a graph $G$ is {\em connected} if $G$ has an edge linking each pair of edges in $M$. The problem to find large connected matchings in graphs $G$ with $\alpha(G)=2$ is closely related to Hadwiger's conjecture for graphs…

An immersion of a graph $H$ into a graph $G$ is a one-to-one mapping $f:V(H) \to V(G)$ and a collection of edge-disjoint paths in $G$, one for each edge of $H$, such that the path $P_{uv}$ corresponding to edge $uv$ has endpoints $f(u)$ and…

The independence polynomial of a graph $G$, denoted $I(G,x)$, is the generating polynomial for the number of independent sets of each size. The roots of $I(G,x)$ are called the \textit{independence roots} of $G$. It is known that for every…

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 Connected Vertex Cover problem is to decide if a graph G has a vertex cover of size at most $k$ that induces a connected subgraph of $G$. This is a well-studied problem, known to be NP-complete for restricted graph classes, and, in…

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent…

We prove that for all $0\leq t\leq k$ and $d\geq 2k$, every graph $G$ with treewidth at most $k$ has a `large' induced subgraph $H$, where $H$ has treewidth at most $t$ and every vertex in $H$ has degree at most $d$ in $G$. The order of $H$…

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem…

Let $\mathcal{H}$ be a class of given graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ contains no induced copies of $H$ for any $H \in \mathcal{H}$. In this article, we characterize all pairs $\{R,S\}$ of graphs such that every…

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent…

A widely open conjecture proposed by Bollob\'as, Erd\H{o}s, and Tuza in the early 1990s states that for any $n$-vertex graph $G$, if the independence number $\alpha(G) = \Omega(n)$, then there is a subset $T \subseteq V(G)$ with $|T| =…

An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and…

Let $G$ and $H$ be simple 3-connected graphs such that $G$ has an $H$-minor. An edge $e$ in $G$ is called {\it $H$-deletable} if $G\backslash e$ is 3-connected and has an $H$-minor. The main result in this paper establishes that, if $G$ has…

We introduce and study the pinnacle sets of a simple graph $G$ with $n$ vertices. Given a bijective vertex labeling $\lambda\,:\,V(G)\rightarrow [n]$, the label $\lambda(v)$ of vertex $v$ is a pinnacle of $(G, \lambda)$ if…

Dallard, Milani\v{c}, and \v{S}torgel [arXiv '22] ask if for every class excluding a fixed planar graph $H$ as an induced minor, Maximum Independent Set can be solved in polynomial time, and show that this is indeed the case when $H$ is any…

A set $X \subseteq V(G)$ in a graph $G$ is $(q,k)$-unbreakable if every separation $(A,B)$ of order at most $k$ in $G$ satisfies $|A \cap X| \leq q$ or $|B \cap X| \leq q$. In this paper, we prove the following result: If a graph $G$…

A graph $G$ contains another graph $H$ as an immersion if $H$ can be obtained from a subgraph of $G$ by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if $G$…

The deficiency of a graph $G$, denoted by $\kd(G)$, is the number of vertices not saturated by a maximum matching. A bone $B_i$ is the tree obtained by attaching two pendent edges to each of the end vertices of a path $P_{i}$. The local…

We prove that for every complete graph $K_t$, all graphs $G$ with no induced subgraph isomorphic to a subdivision of $K_t$ have a stable subset of size at least $|G|/{\rm polylog}|G|$. This is close to best possible, because for $t\ge 7$,…

We prove that every graph $G$ on $n$ vertices with no isolated vertices contains an induced subgraph of size at least $n/10000$ with all degrees odd. This solves an old and well-known conjecture in graph theory.