Algorithmic and combinatorial aspects of graph monitoring and defense problems

Abstract

Many real world optimization problems can be modeled as graph optimization problems. However, several graph optimization problems that are practically significant are NP-hard for general graphs, and hence, it is unlikely to find exact solutions in polynomial-time. One approach to tackle this is to study the problem for some restricted graph classes, as most of the time graphs obtained by modeling real world problems exhibit some special properties. Many of the researchers are working to design polynomial-time algorithms for NP-hard graph optimization problems for restricted graph classes. This thesis considers the following important graph optimization problems, namely (i) ET E R NA L VE RT E X COV E R Problem, (ii) CO N N E C T E D (& ET E R NA L CO N N E C T E D ) VE RT E X COV E R Problem, (iii) ( I N D E P E N D E N T ) ROMA N DOMI NAT I O N Problem, (iv) WE A K ROMA N DOMI NAT I O N Problem and (v) MO N I TO R I N G ED G E -GE O D E T I C S E T Problem. Given a graph G = (V,E), a set S of vertices is said to be a vertex cover if for every edge e ∈ E, S contains at least one endpoint of S. The eternal vertex cover problem is a variant of the vertex cover problem. It is a two-player (attacker and defender) game in which, given a graph G = (V,E), the defender needs to allocate guards at some vertices so that the allocated vertices form a vertex cover. The attacker can attack one edge at a time, and the defender needs to move the guards along the edges such that at least one guard moves through the attacked edge and the new configuration still remains a vertex cover. The attacker wins if no such move exists for the defender. The defender wins if there exists a strategy to defend the graph against any infinite sequence of attacks. The minimum number of guards with which the defender can form a winning strategy is called the eternal vertex cover number of G, and is denoted by evc(G). Given a graph G, the problem of finding evc(G) is said to be the ET E R NA L VE RT E X COV E R Problem. For some special graph classes, we provide polynomial-time algorithms to solve the ET E R NA L VE RT E X COV E R Problem. Given a graph G = (V,E), a vertex cover S ⊆ V is called a connected vertex cover if the graph induced on S is connected. Given a graph G = (V,E), the problem of finding cvc(G) is said to be the CO N N E C T E D VE RT E X COV E R Problem. The ET E R NA L VE RT E X COV E R Problem is referred to as the ET E R NA L CO N N E C T E D VE RT E X COV E R Problem if the following additional condition is added: underlying vertices of each defensive configuration form a connected vertex cover. The smallest number of guards that can be used to create a successful defensive strategy, in this case, is known as the eternal connected vertex cover number of G and is denoted by the ecvc(G). We provide polynomial-time algorithms to solve the CO N N E C T E D VE RT E X COV E R Problem and ET E R NA L CO N N E C T E D VE RT E X COV E R Problem for some restricted graph classes. We also show that ET E R NA L CO N N E C T E D VE RT E X COV E R Problem is NP-hard for Hamiltonian graphs. Given a graph G = (V,E), a function f : V → {0, 1, 2} is said to be a Roman Dominating function if for every v ∈ V with f(v) = 0, there exists a vertex u ∈ N(v) such that f(u) = 2. A Roman Dominating function f is said to be an Independent Roman Dominating function (or IRDF), if V1 ∪ V2 forms an independent set, where Vi = {v ∈ V | f(v) = i}, for i ∈ {0, 1, 2}. The total weight of f is equal to P v∈V f(v), and is denoted as w(f). The Roman domination number (resp. Independent Roman domination number) of G, denoted by γR(G) (resp. iR(G)), is defined as min{w(f) | f is an RDF (resp. IRDF) of G}. For a given graph G, the problem of computing γR(G) (resp. iR(G)) is defined as the ROMA N DOMI NAT I O N Problem (resp. I N D E P E N D E N T ROMA N DOMI NAT I O N Problem). We designed polynomial-time algorithms to solve the I N D E P E N D E N T ROMA N DOMI NAT I O N Problem for some restricted graph classes. We also provide some parameterized complexity results for the ( I N D E P E N D E N T ) ROMA N DOMI NAT I O N Problem. Consider a graph G = (V,E) and a function f : V → {0, 1, 2}. A vertex u with f(u) = 0 is defined as undefended by f if it lacks adjacency to any vertex with a positive f-value. The function f is said to be a Weak Roman Dominating function (or WRD function) if, for every vertex u with f(u) = 0, there exists a neighbour v of u with f(v) > 0, such that f′ : V → {0, 1, 2} defined in the following way: f′(u) = 1, f′(v) = f(v) − 1, and f′(w) = f(w), for all vertices w in V \ {u, v}; so that no vertices are undefended by f′. The total weight of f is equal to P v∈V f(v), and is denoted as w(f). The Weak Roman domination number denoted by γr(G), represents min{w(f) | f is a WRD function of G}. For a given graph G, the problem of finding γr(G) is defined as the WE A K ROMA N DOMI NAT I O N Problem. We present some algorithmic (and hardness) results for some restricted graph classes. We also provide some approximation results for this problem. Given a graph G = (V,E), a set S ⊆ V is said to be a monitoring edge-geodetic set if the deletion of any edge in the graph results in a change in the distance between at least one pair of vertices in S. The minimum size of such a set in G is called the monitoring edge-geodetic number of G and is denoted by meg(G). For a given graph G, the problem of finding meg(G) is defined as the MO N I TO R I N G ED G E -GE O D E T I C S E T Problem. In our work, we provide efficient algorithms to compute the monitoring edge-geodetic number efficiently for some special graph classes, which follow from structural characterizations of the optimal monitoring edge-geodetic sets for these graph classes.