Related papers: $k$-Fair Coalitions in Graphs

A set of vertices in a graph is agile if, however we partition the set into two parts, we can always find two vertex-disjoint connected subgraphs where one covers the first and the other the second part. We present a characterization for…

A dominating set of a graph $G$ is a set $S \subseteq V(G)$ such that every vertex in $V(G) \setminus S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. A secure dominating set of $G$ is a dominating set $S$ of…

Given a graph $G=(V(G), E(G))$, the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph $G$ are denoted by $\gamma(G)$, $\gamma_{\rm pr}(G)$, and $\gamma_{t}(G)$, respectively. For…

Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $\gamma_{t}$($G$), is the minimum cardinality of a total…

Many approximation algorithms and heuristic algorithms to find a fair clustering have emerged. In this paper we define a new and natural variant of fair clustering problem and design a polynomial time algorithm to compute an optimal fair…

Let $G=(V,E)$ be a simple graph without isolated vertices. A set $S\subseteq V$ is a paired-dominating set if every vertex in $V-S$ has at least one neighbor in $S$ and the subgraph induced by $S$ contains a perfect matching. In this paper,…

We study the fair k-set selection problem where we aim to select $k$ sets from a given set system such that the (weighted) occurrence times that each element appears in these $k$ selected sets are balanced, i.e., the maximum (weighted)…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex that does not belong to $S$ is adjacent to a vertex in $S$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. The…

An equitable partition of a graph $G$ is a partition of the vertex-set of $G$ such that the sizes of any two parts differ by at most one. We show that every graph with an acyclic coloring with at most $k$ colors can be equitably partitioned…

Let $F_1, F_2, ..., F_k$ be graphs with the same vertex set $V$. A subset $S \subseteq V$ is a simultaneous dominating set if for every $i$, $1 \le i \le k$, every vertex of $F_i$ not in $S$ is adjacent to a vertex in $S$ in $F_i$; that is,…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

A (not necessarily proper) vertex coloring of a graph $G$ with color classes $V_1$, $V_2$, $\dots$, $V_k$, is said to be a {\it Fair And Tolerant vertex coloring of $G$ with $k$ colors}, whenever $V_1$, $V_2$, $\dots$, $V_k$ are nonempty…

A path in an(a) edge(vertex)-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called \emph{conflict-free (vertex-)connected} if there is…

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have…

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors…

An unfriendly partition of a graph $G = (V,E)$ is a function $c: V \to 2$ such that $|\{x\in N(v): c(x)\neq c(v)\}|\geq |\{x\in N(v): c(x)=c(v)\}|$ for every vertex $v\in V$, where $N(v)$ denotes its neighborhood. It was conjectured by…

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $G$ is in $S$ or is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The domination number…

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…

Let $F_k$ be the set of graphs on $k$ vertices. For a graph $G$, a $k$-decomposition is a set of induced subgraphs of $G$, each isomorphic to an element of $F_k$, such that each pair of vertices of $G$ is in exactly one element of the set.…

For a hypergraph $H$ let $\beta(H)$ denote the minimal number of edges from $H$ covering $V(H)$. An edge $S$ of $H$ is said to represent {\em fairly} (resp. {\em almost fairly}) a partition $(V_1,V_2, \ldots, V_m)$ of $V(H)$ if $|S\cap…