Related papers: Exactly Hittable Interval Graphs

The Graph Minors Structure Theorem of Robertson and Seymour asserts that, for every graph $H,$ every $H$-minor-free graph can be obtained by clique-sums of ``almost embeddable'' graphs. Here a graph is ``almost embeddable'' if it can be…

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a…

A graph is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G, there exists a unique smallest Helly graph H(G) into which G…

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 is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated theorem of Dirac from 1952 asserts that every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to…

A graph $G$ is {\em matching-decyclable} if it has a matching $M$ such that $G-M$ is acyclic. Deciding whether $G$ is matching-decyclable is an NP-complete problem even if $G$ is 2-connected, planar, and subcubic. In this work we present…

Intersection graphs of planar geometric objects such as intervals, disks, rectangles and pseudo-disks are well studied. Motivated by various applications, Butman et al. in SODA 2007 considered algorithmic questions in intersection graphs of…

The identifiability problem arises naturally in a number of contexts in mathematics and computer science. Specific instances include local or global rigidity of graphs and unique completability of partially-filled tensors subject to rank…

A dominating set $S$ in a graph $G$ is said to be perfect if every vertex of $G$ not in $S$ is adjacent to just one vertex of $S$. Given a vertex subset $S'$ of a side $P_m$ of an $m\times n$ grid graph $G$, the perfect dominating sets $S$…

A subset \( C \) of the vertex set \( V \) of a graph \( \Gamma = (V,E) \) is termed an $(r,s)$-regular set if each vertex in \( C \) is adjacent to exactly \( r \) other vertices in \( C \), while each vertex not in \( C \) is adjacent to…

For a fixed set ${\cal H}$ of graphs, a graph $G$ is ${\cal H}$-subgraph-free if $G$ does not contain any $H \in {\cal H}$ as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on ${\cal…

A cluster graph is a graph whose every connected component is a complete graph. Given a simple undirected graph $G$, a subset of vertices inducing a cluster graph is called an independent union of cliques (IUC), and the IUC polytope…

A graph $G$ is called \emph{symmetric with respect to a functional $F_G(P)$} defined on the set of all the probability distributions on its vertex set if the distribution $P^*$ maximizing $F_G(P)$ is uniform on $V(G)$. Using the…

We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph $H$ and an input graph $G$, the task is to find a set $X \subseteq V(G)$ of minimum size that hits all subgraphs of $G$ isomorphic to…

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a…

We study the parameterized complexity of dominating sets in geometric intersection graphs. In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a…

Let $G$ be a group. The intersection subgroup graph of $G$ (introduced by Anderson et al. \cite{anderson}) is the simple graph $\Gamma_{S}(G)$ whose vertices are those non-trivial subgroups say $H$ of $G$ with $H\cap K=\{e\}$ for some…

For a graph whose vertex set is a finite set of points in the Euclidean $d$-space consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect.…

A digraph consisting of a set of vertices $V$ and a set of arcs $E$ is called an interval digraph if there exists a family of closed intervals $\{I_u,J_u\}_{u \in V}$ such that $uv$ is an arc if and only if the intersection of $I_u$ and…

A group $G$ is integrable if it is isomorphic to the derived subgroup of a group $H$; that is, if $H'\simeq G$, and in this case $H$ is an integral of $G$. If $G$ is a subgroup of $U$, we say that $G$ is integrable within $U$ if $G=H'$ for…