Abstract
In this paper, we introduce a new representation of simple undirected graphs in terms of set of vectors in finite dimensional vector spaces over \(\mathbb {Z}_2\) which satisfy consecutive 1’s property, called a coding sequence of a graph G. Among all coding sequences we identify the one which is unique for a class of isomorphic graphs, called the code of a graph. We characterize several classes of graphs in terms of coding sequences. It is shown that a graph G with n vertices is a tree if and only if any coding sequence of G is a basis of the vector space \(\mathbb {Z}_2^{n-1}\) over \(\mathbb {Z}_2\).
Moreover, considering coding sequences as binary matroids, we obtain a characterization for simple graphic matroids. Introducing concepts of segment binary matroid and strong isomorphisms we show that two simple undirected graphs are isomorphic if and only if their canonical sequences are strongly isomorphic simple segment binary matroids.
AMS Subject Classifications: 05C62, 05C50, 05B35.
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Acknowledgements
Authors are thankful to learned referees for their kind comments and suggestions. The second author is grateful to the University Grants Commission, Government of India, for providing research support (Grant no. F. 17-76/2008(SA-I).
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Ghosh, S., Sen Gupta, R., Sen, M.K. (2016). An Introduction to Coding Sequences of Graphs. In: Chan, TH., Li, M., Wang, L. (eds) Combinatorial Optimization and Applications. COCOA 2016. Lecture Notes in Computer Science(), vol 10043. Springer, Cham. https://doi.org/10.1007/978-3-319-48749-6_15
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DOI: https://doi.org/10.1007/978-3-319-48749-6_15
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