Related papers: Partitioning a Graph into Disjoint Cliques and a T…

This paper investigates the computational complexity of deciding whether the vertices of a graph can be partitioned into a disjoint union of cliques and a triangle-free subgraph. This problem is known to be $\NP$-complete on arbitrary…

Motivated by Chudnovsky's structure theorem of bull-free graphs, Abu-Khzam, Feghali, and M\"uller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a triangle-free graph is NP-complete for five…

A graph $G$ is said to be $2$-divisible if for all (nonempty) induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $\omega(A) < \omega(H)$ and $\omega(B) < \omega(H)$. A graph $G$ is said to be perfectly…

A graph $G$ has a perfect division if its vertex set can be partitioned into two sets $A$, $B$ such that $G[A]$ is perfect and $\omega(G[B]) < \omega(G)$. We call $G$ perfectly divisible if every induced subgraph of $G$ admits a perfect…

For a graph $G$ and a set of graphs $\mathcal{H}$, we say that $G$ is {\em $\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\mathcal{F}$,…

We show that the following problems are NP-complete. 1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph? 2. Is the difference between the chromatic number and clique number at most $1$…

A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$…

A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$, and a graph $G$ is perfectly weight divisible if for every…

A graph $G=(V,E)$ is a {\it unipolar graph} if there exits a partition $V=V_1 \cup V_2$ such that, $V_1$ is a clique and $V_2$ induces the disjoint union of cliques. The complement-closed class of {\it generalized split graphs} are those…

Let $G = (V, E)$ be a connected graph with maximum degree $k\geq 3$ distinct from $K_{k+1}$. Given integers $s \geq 2$ and $p_1,\ldots,p_s\geq 0$, $G$ is said to be $(p_1, \dots, p_s)$-partitionable if there exists a partition of $V$ into…

For a connected graph G=(V,E), a subset U of V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is…

A graph $G$ of order $nv$ where $n\geq 2$ and $v\geq 2$ is said to be weakly $(n,v)$-clique-partitioned if its vertex set can be decomposed in a unique way into $n$ vertex-disjoint $v$-cliques. It is strongly $(n,v)$-clique-partitioned if…

Let $S=\{K_{1,3},K_3,P_4\}$ be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph $G$ into graphs taken from any non-empty $S'\subseteq S$. The problem is known to be NP-complete for any…

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. It is easy…

An $({\cal F},{\cal F}_d)$-partition of a graph is a vertex-partition into two sets $F$ and $F_d$ such that the graph induced by $F$ is a forest and the one induced by $F_d$ is a forest with maximum degree at most $d$. We prove that every…

A graph $G$ is perfectly divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $\omega(H[X_1]) < \omega(H)$ and $H[X_2]$ is a perfect graph. In this…

We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erd\H{o}s' Sparse Half Conjecture. Among others we prove the following variant of it: For every sufficiently large even integer $n$ the…

A graph $G$ is {\em $k$-choosable} if for every assignment of a set $S(v)$ of $k$ colors to every vertex $v$ of $G$, there is a proper coloring of $G$ that assigns to each vertex $v$ a color from $S(v)$. We consider the complexity of…

A graph $G$ is {\em perfectly divisible} if, for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. A {\em bull} is a graph consisting of a triangle with…

An $(r, \ell)$-partition of a graph $G$ is a partition of its vertex set into $r$ independent sets and $\ell$ cliques. A graph is $(r, \ell)$ if it admits an $(r, \ell)$-partition. A graph is well-covered if every maximal independent set is…