Numerous applications of resource partitioning are found in electrical power distribution systems, telecommunication networks, computer networks, fault tolerant systems, grid computing etc. The resource partitioning problem is concerned with finding a resource k-partition of a graph. In this book we present linear algorithms to compute a resource tripartition of a triconnected planar graph and a resource 4-partition of a 4-connected planar graph with base vertices located on the same face of a planar embedding. To solve the resource tripartitioning problem, we have developed a linear- time algorithm for constructing a nonseparating ear decomposition through two vertices a,b and avoiding a third vertex c of a 3-connected planar graph for any three vertices a,b,c. We also give bounds on the number of ears and the length of an ear for the nonseparating ear decomposition produced by our algorithm. This book will especially be useful to professionals in Algorithms and Graph Theory.