Weakly modular graphs are a well-studied graph class in metric graph theory. Here we extend the concept to directed graphs in such a way that symmetric (undirected) case is included and key properties, in particular closure under Cartesian products and gated amalgams continue to be satisfied. We characterize directed weakly modular graphs in terms of their directed interval functions. We then consider extensions to the directed setting for two important subclasses: modular and median graphs.
Permutation codes under the generalized Kendall- τ metric are error-correcting codes with applications in rank modulation and related storage systems. Constructing permutation arrays is a challenging problem since even computing the distance between two permutations is NP-hard. This paper investigates (n, d)-permutation arrays (PAs) with prescribed length of permutations n and minimum generalized Kendall- τ distance d. We determine exact values of T(n, d), the maximum size of a PA, for n=4 and d=2,3 . We develop algorithms for computing Kendall– τ distances and constructing permutation arrays for n ≤ 11 . An efficient testing procedure for (n, 2)-PAs is also presented.
The ℱ -minor-free-deletion problem asks to find minimum number of vertices S in a given graph G such that G ∖ S does not contain any graph from the specified family ℱ as a minor. In this paper, we consider {K_2,3,K_4} -minor-free-deletion problem in AT-free graphs, which is equivalent to finding a maximum induced outer-planar graph of a given AT-free graph. We show that this problem is solvable in polynomial-time. This problem can also be seen as a natural extension of minimum vertex cover and minimum feedback vertex set problem since they are equivalent to {K_2} and {K_3} minor free deletion problems respectively and are known to be polynomial time solvable in AT-free graphs [2, 24]. Additionally, we provide a polynomial-time algorithm for finding a maximum induced linear forest in AT-free graphs, which serves as a key subroutine in our algorithm to compute a maximum induced outer-planar graph.
A matching M in a graph G is said to be extendable to a perfect matching if there exists a perfect matching M^* of G such that M ⊆ M^* . In this work, we study the extendability of matchings under a neighbourhood condition: no unsaturated vertex has all of its neighbours M-saturated. Vandenbussche and West showed that, in the hypercube Q_n , any matching of size at most 2n - 4 is extendable to a perfect matching if and only if it satisfies this condition. We extend their result to the Cartesian product Q_n □ P_m by proving that every matching of size at most 2n - 2 is extendable to a perfect matching if and only if it does not saturate the neighbourhood of any unsaturated vertex. Furthermore, we demonstrate that this bound is tight by constructing a matching of size 2n + 2δ (H) - 3 in Q_n □ H that satisfies the neighbourhood condition but is not extendable to a perfect matching.
Let G=(V, E) be a simple undirected graph with no isolated vertex. A set D_t⊆ V is a total dominating set of G if (i) D_t is a dominating set, and (ii) the set D_t induces a subgraph with no isolated vertex. The total dominating set of the minimum cardinality is called the minimum total dominating set, and the size of the minimum total dominating set is called the total domination number ( γ _t(G) ). Given a graph G, the total dominating set (TDS) problem is to find a total dominating set of minimum cardinality. In this paper, we enhance the hardness of the TDS problem by proving that it is NP-complete on grid-aligned UDGs, a subclass of unit disk graphs (UDGs). Furthermore, we present a 6.29 factor approximation algorithm for the TDS problem in UDGs.
