Graphs with bounded induced distance

Networks with small stretch number

“…Finally, we give a characterization result in term of forbidden subgraphs for the class BID(2 − 1/i), for each integer i > 1. This result has been obtained by extending the technique used in [7] to show a similar characterization for the class BID(3/2). In turn, this new result allows us to design a polynomial time algorithm to solve the recognition problem for the class BID(2 − 1/i), for each i 1 (if k is not fixed, this problem is Co-NP-complete for the class BID(k) [7]).…”

“…This result has been obtained by extending the technique used in [7] to show a similar characterization for the class BID(3/2). In turn, this new result allows us to design a polynomial time algorithm to solve the recognition problem for the class BID(2 − 1/i), for each i 1 (if k is not fixed, this problem is Co-NP-complete for the class BID(k) [7]). Unfortunately, the running time of this algorithm is exponential in i (more precisely, it is bounded by O(n 3i+2 )).…”

“…In this context we show that: (i) there is no graph G with stretch number s(G) such that 2 − 1/i < s(G) < 2 − 1/(i + 1), for each integer i 1 (this fact was conjectured in [7]); (ii) there exists a graph G such that s(G) = 2 − 1/i, for each integer i 1. These results give a partial solution to the following more general problem: Given a rational number k, is k an admissible stretch number, i.e., is there a graph G such that s(G) = k?…”

“…We model a network in which node faults have occurred by the subnetwork induced by the non-faulty components. Using this model, in [7] we have introduced the class BID(k) of graphs with bounded induced distance of order k. A network modeled as a graph in BID(k) can be characterized as follows: if some nodes have failed, as long as two nodes remain connected, the distance between these nodes in the faulty graph is at most k times the distance in the non-faulty graph.…”

“…Some characterization, complexity, and structural results about BID(k) are given in [7]. In particular, the concept of stretch number has been introduced: the stretch number s(G) of a given graph G is the smallest rational number k such that G belongs to BID(k).…”