Reconstruction of interval graphs

Authors:
Masashi Kiyomi;Toshiki Saitoh;Ryuhei Uehara
Affiliations:
School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan;ERATO MINATO Discrete Structure Manipulation System Project, JST, North 14, West 9, Sapporo, Hokkaido 060-0814, Japan;School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan
Venue:
Theoretical Computer Science
Year:
2010

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Abstract

The graph reconstruction conjecture is a long-standing open problem in graph theory. There are many algorithmic studies related to it, such as DECK CHECKING, LEGITIMATE DECK, PREIMAGE CONSTRUCTION, and PREIMAGE COUNTING. We study these algorithmic problems by limiting the graph class to interval graphs. Since we can solve GRAPH ISOMORPHISM for interval graphs in polynomial time, DECK CHECKING for interval graphs is easily done in polynomial time. Since the number of interval graphs that can be obtained from an interval graph by adding a vertex and edges incident to it can be exponentially large, developing polynomial time algorithms for LEGITIMATE DECK, PREIMAGE CONSTRUCTION, and PREIMAGE COUNTING on interval graphs is not trivial. We present that these three problems are solvable in polynomial time on interval graphs.