DeepAI AI Chat
Log In Sign Up

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

03/14/2021
by   Márcia R. Cappelle, et al.
0

Let G=(V(G),E(G)) be a finite simple undirected graph with vertex set V(G), edge set E(G) and vertex subset S⊆ V(G). S is termed open-dominating if every vertex of G has at least one neighbor in S, and open-independent, open-locating-dominating (an OLD_oind-set for short) if no two vertices in G have the same set of neighbors in S, and each vertex in S is open-dominated exactly once by S. The problem of deciding whether or not G has an OLD_oind-set has important applications, has been studied elsewhere and is known to be 𝒩𝒫-complete. Reinforcing this result, it appears that the problem is notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs. Also, we present characterizations of both P_4-tidy graphs and the complementary prisms of cographs, that have an OLD_oind-set.

READ FULL TEXT

page 1

page 2

page 3

page 4

08/10/2020

Finding Efficient Domination for S_1,3,3-Free Bipartite Graphs in Polynomial Time

A vertex set D in a finite undirected graph G is an efficient dominating...
02/04/2023

Extremal digraphs for open neighbourhood location-domination and identifying codes

A set S of vertices of a digraph D is called an open neighbourhood locat...
08/18/2022

Recognizing well-dominated graphs is coNP-complete

A graph G is well-covered if every minimal vertex cover of G is minimum,...
11/06/2018

Finding Independent Transversals Efficiently

We give an efficient algorithm that, given a graph G and a partition V_1...
09/05/2021

A linear-time algorithm for semitotal domination in strongly chordal graphs

In a graph G=(V,E) with no isolated vertex, a dominating set D ⊆ V, is c...
03/01/2023

Scarf's algorithm and stable marriages

Scarf's algorithm gives a pivoting procedure to find a special vertex – ...
05/15/2019

Perfect Italian domination on planar and regular graphs

A perfect Italian dominating function of a graph G=(V,E) is a function f...