Rainbow Connectivity in Graphs

1. Rainbow Connectivity in Graphs 01st April 2012 Karthik C S IIT Bombay

2. Rainbow Connectivity • A graph is a collection of vertices or nodes and a collection of edges that connect pairs of vertices. • A path in an edge coloured graph is said to be a rainbow path if no two edges on the path have the same colour. • An edge coloured graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. • The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colours required to edge colour the graph such that G is (strongly) rainbow connected.

3. Examples 6 5 5 4 4 3 3 1 2 1 2 3-rainbow connected 2-rainbow connected

4. Question • Given a graph G, how hard is it to determine if rc(G)<k, for some constant k? • Given a graph G, how hard is it to determine if src(G)<k, for some constant k? Rainbow Connectivity: Hardness and Tractability by P. Ananth, M. Nasre, and K.K. Sarpatwar. (http://drops.dagstuhl.de/opus/volltexte/2011/3353/pdf/35.pdf)

5. Hardness • How do we talk about the hardness of a particular problem? • If we know about certain problems which are pre-classified as hard to solve then by establishing a reduction from one of these known hard problems to our problem we can say that even solving our problem is hard if not harder (as long as the reduction takes very little time to execute - say polynomial in nature). • So our aim now is given an instance of a well known hard (NP Hard) problem we would like to construct an instance of our problem such that solving our problem would solve the known hard problem we have taken.

6. Vertex colouring problem • Given a graph G we would like to determine if it is possible to use less than or equal to k ( k > 3 ) colours to colour all the vertices such that no two adjacent vertices are of the same colour. • This problem is classified as NP Hard or in simple words very hard to solve. • Now the choice of this problem for our reduction is more or less obvious as we are dealing with problems which involve colouring of graphs.

7. The Reduction Instance of Vertex Colouring Problem ? Construct Instance of src(G)<k Problem Send SOLVER Determines if src(G)<k ? Interpret Exactly answer above instance of Vertex Colouring Problem

8. Alternate way out Vertex colouring Problem Vertex colouring Problem Direct Reduction Reduction Strong Rainbow Connectivity Problem Problem “X” Reduction Looks hard! Strong Rainbow Connectivity Problem

9. Problem X • In strong rainbow connectivity problem we have to check for every vertex pair if there exists a shortest path which is also a rainbow path using k or less colours. • So we look at addressing a somewhat easier problem. Instead of checking for every vertex pair we check for only a subset of these vertex pairs. • Thinking of this intermediate problem is fairly non trivial but a common practice in computer science.

10. Subset Strong Rainbow Connectivity Problem Given a graph G and subset P of all the vertex pairs in G, how hard is it to determine if src(P)< k, for some constant k? In other words how hard is to determine if we can colour the edges in G using k such that for every vertex pair in P there exists a shortest path which is also a rainbow path.

11. Reduction Part 1 Subset Strong Rainbow Connectivity Problem Vertex Colouring Problem Vertices in G are edges in H • G • H n+1 vertices n edges n vertices

12. Reduction Part 1 Subset Strong Rainbow Connectivity Problem Vertex Colouring Problem Vertices in G are edges in H • G • H Here with minimum colours we have to colour vertices such that no two adjacent vertices have the same colour Here with minimum colours we have to colour edges such that for every vertex pair in P there is a geodesic rainbow path

13. Reduction Part 1 Subset Strong Rainbow Connectivity Problem Vertex Colouring Problem Vertices in G are edges in H • G • H Let every vertex in G correspond to an edge in H. Then the shortest path between any two vertices in H are always two edges which are adjacent.

14. Reduction Part 1 Subset Strong Rainbow Connectivity Problem Vertex Colouring Problem Vertices in G are edges in H • G • H So if two vertices are connected by an edge in G then we include the pair in P. This pair in H can have a shortest path which is a rainbow path only if the 2 edges connecting them are of different colour which is precisely what we need to resolve vertex colouring problem in G.

15. Reduction Part 2 Subset Strong Rainbow Connectivity Problem Strong Rainbow Connectivity Problem Add edges (and vertices?) to make geodesic rainbow paths for all vertex pairs not in P • H • G’ Then solving G’ for strong rainbow connectivity problem would solve H for subset strong rainbow connectivity problem because colouring these new edges will not affect vertex pairs in P.

16. Reduction Part 2 • H Consider all vertex pairs in G which are not present in P • W • H Now to establish a path(should be of length 2 or less) between them we introduces set of vertices Wand for every vertex pair not in P we connect them through an unique vertex in W. Vertex pairs to take care of: All vertex pairs in G not in P

17. Reduction Part 2 • H Consider all vertex pairs in W We take care of these by considering another copy of W, say W’ and forming a completely bipartite graph between them • W • H Vertex pairs to take care of: All vertex pairs in W All vertices W and central vertex of H All vertices of W and H which do not share an edge

18. Reduction Part 2 Perfect Match Colouring • H Now to resolve for all vertices in W, W’and central vertex of H, we connect all vertices in W’ to central vertex in H • W’ • W • H • Vertex pairs to take care of: • All vertices W,W’ and central vertex of H • All vertices of W and H which do not share an edge • All vertices of W’ and H which do not share an edge

19. Reduction Part 2 • H • W • W’ • H Now this part might be slightly hard to grasp. Since we want to establish connections outside the star, we make copies of all non central vertices of H, say H’ to connect them to W and W’. • Vertex pairs to take care of: • All vertices of W and H which do not share an edge • All vertices of W’ and H which do not share an edge

20. Reduction Part 2 • W’ • W • H We have only vertices of H’ and central vertex of H left. This we can take care of as we did for W by constructing a complete bipartite graph H’’. This should also be edge connected to W. Perfect Match Colouring • H’ Vertex pairs to take care of: All vertices of H’ and central vertex of H

21. Reduction Part 2 • W’ • W • H Regarding the colouring of the new edges, say we have 3 colours Red, Blue and Green. We colour the edges involved in complete bipartite graphs as follows: Perfect matching – Blue 2. Remaining Edges – Green The edges between H and H’, H’’, W’ are coloured using Red. • H’ The edges between H and W are colored Blue and Green pairwise. • H’’ • G’

22. Reduction Part 2 • W’ • W • H We have taken care of every vertex pair in the constructed graph except those in P. Thus solving G’ for strong rainbow connectivity problem will solve subset rainbow connectivity problem in H. • H’ • H’’ • G’

23. Complete Reduction • G • H • G’

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