Network Coding encourages information coding across a communication network. While the necessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusing on algebraic or information theoretic aspects of the problem. This work aims at an in-depth examination of the relation between algebraic coding and network topologies. We mathematically establish a series

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... of results along the direction of: if network coding is necessary/beneficial, or if a particular finite field is required for coding, then the network must have a corresponding hidden structure embedded in its underlying topology, and such embedding is computationally efficient to verify. Specifically, we first formulate a meta-conjecture, the NC-Minor Conjecture, that articulates such a connection between graph theory and network coding, in the language of graph minors. We next prove that the NC-Minor Conjecture is almost equivalent to the Hadwiger Conjecture, which connects graph minors with graph coloring. Such equivalence implies the existence of K4, K5, K6, and K O(q/ log q) minors, for networks requiring F3, F4, F5 and Fq, respectively. We finally prove that network coding can make a difference from routing only if the network contains a K4 minor, and this minor containment result is tight. Practical implications of the above results are discussed. ) is the corresponding author. During the past twelve years, a plethora of results have been obtained on the theory of network coding, leading to advanced understandings of the subject, especially for the single source case. Existing work usually approaches network coding from an algebraic or information theoretic perspective, and treats the graph topology of a network as a black box. Latest results suggest that a close examination of the network structure and exploiting in-depth connections between graph theory and network coding may lead to new understandings on when and how network coding should be performed. For example, while previous research suggest that the necessary field size grows with the number of receivers and has no finite bound [4], coding over very small finite fields suffices for networks exhibiting a planar or close-to-planar topology [5] .