Related papers: About $(k,l)$-kernels, semikernels and Grundy func…

An independent vertex subset $S$ of the directed graph $G$ is a kernel if the set of out-neighbors of $S$ is $V(G)\setminus S$. An independent vertex subset $Q$ of $G$ is a quasi-kernel if the union of the first and second out-neighbors…

The Grundy number of a graph $G$, denoted by $\Gamma(G)$, is the largest $k$ such that there exists a partition of $V(G)$, into $k$ independent sets $V_1,\ldots, V_k$ and every vertex of $V_i$ is adjacent to at least one vertex in $V_j$,…

We present a polyhedral description of kernels in orientations of line multigraphs. Given a digraph $D$, let $FK(D)$ denote the fractional kernel polytope defined on $D$, and let ${\sigma}(D)$ denote the linear system defining $FK(D)$. A…

For a given graph $G = (V, E)$, a subset of the vertices $D\subseteq V$ is called a semitotal dominating set, if $D$ is a dominating set and every vertex $v \in D$ is within distance two to another witness $v' \in D$. We want to find a…

For a fixed graph $H$, in the List $H$-Coloring problem, we are given a graph $G$ along with list $L(v) \subseteq V(H)$ for every $v \in V(G)$, and we have to determine if there exists a list homomorphism $\varphi$ from $(G,L)$ to $H$,…

Any directed graph $D=(V(D),A(D))$ in this work is assumed to be finite and without self-loops. A source in a directed graph is a vertex having at least one ingoing arc. A quasi-kernel $Q\subseteq V(D)$ is an independent set in $D$ such…

A graph is $(d_1, ..., d_r)$-colorable if its vertex set can be partitioned into $r$ sets $V_1, ..., V_r$ so that the maximum degree of the graph induced by $V_i$ is at most $d_i$ for each $i\in \{1, ..., r\}$. For a given pair $(g, d_1)$,…

The fractional arboricity of a digraph $D$, denoted by $\gamma(D)$, is defined as $\gamma(D)= \max_{H \subseteq D, |V(H)| >1} \frac {|A(H)|} {|V(H)|-1}$. Frank in [Covering branching, Acta Scientiarum Mathematicarum (Szeged) 41 (1979),…

A coloring of a graph G = (V,E) is a partition {V1, V2, . . ., Vk} of V into independent sets or color classes. A vertex v Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj . A coloring is a Grundy…

A digraph is strongly connected if it has a directed path from $x$ to $y$ for every ordered pair of distinct vertices $x, y$ and it is strongly $k$-connected if it has at least $k+1$ vertices and remains strongly connected when we delete…

A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on…

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…

An acyclic digraph in which every vertex has indegree at most $i$ and outdegree at most $j$ is called an $(i,j)$ digraph for some positive integers $i$ and $j$. The phylogeny graph of a digraph $D$ has $V(D)$ as the vertex set and an edge…

A $(\delta\geq k_1,\delta\geq k_2)$-partition of a graph $G$ is a vertex-partition $(V_1,V_2)$ of $G$ satisfying that $\delta(G[V_i])\geq k_i$ for $i=1,2$. We determine, for all positive integers $k_1,k_2$, the complexity of deciding…

A number of years ago, Kumar Murty pointed out to me that the computation of the fundamental group of a Hilbert modular surface ([7],IV,${\S}$6), and the computation of the congruence subgroup kernel of SL(2) ([6]) were surprisingly…

A subset $S \subseteq V$ in a graph $G = (V,E)$ is called a $[1, k]$-set, if for every vertex $v \in V \setminus S$, $1 \leq | N_G(v) \cap S | \leq k$. The $[1,k]$-domination number of $G$, denoted by $\gamma_{[1, k]}(G)$ is the size of the…

Let $p$ and $q$ be distinct primes. The \textit{semiprime divisor function graph} denoted by $G_{D(pq)}$, is the graph with vertex set $V(G_{D(pq)})=\{1,p,q,pq\}$ and edge set $E(G_{D(pq)})=\{\{1,p\}, \{1,q\},\{1,pq\},\{p,pq\},\{q,pq\}\}$.…

A hypergraph G with n vertices and m hyperedges with d endpoints each is (k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a linearly…

Given any digraph $D$ without loops or multiple arcs, there is a natural construction of a semigroup $\langle D\rangle$ of transformations. To every arc $(a,b)$ of $D$ is associated the idempotent transformation $(a\to b)$ mapping $a$ to…

An induced subgraph is called an induced matching if each vertex is a degree-1 vertex in the subgraph. The \textsc{Almost Induced Matching} problem asks whether we can delete at most $k$ vertices from the input graph such that the remaining…