We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex Ú are as evenly distributed to the left and right of Ú as possible. This problem, which has applications in graph drawing for example, is shown to be NP-hard, and remains NP-hard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertex-ordering, obtaining optimal orderings for directed acyclic graphs and graphs with maximum degree three. Finally we consider the problem of determining a balanced vertex-ordering of a bipartite graph with a fixed ordering of one bipartition. When only the imbalances of the fixed vertices count, this problem is shown to be NP-hard. On the other hand, we describe an optimal linear time algorithm when the final imbalances of all vertices count. We obtain a linear time algorithm to compute an optimal vertex-ordering of a bipartite graph with one bipartition of constant size.
Biedl, Chan, Ganjali, Hajiaghayi, and Wood give a 8 -approximation algorithm for finding an ordering of the vertices such that for each vertex v, the neighbors of v are as evenly distributed to the right and left of v as possible REF .
Balanced vertex-orderings of graphs
Discrete Applied Mathematics
Mathematics Computer Science