spanningvalmaxG

SpanningvalmaxG is a graph-theoretic quantity defined for a finite graph G = (V, E). It denotes the maximum value of a given valuation function f taken over all spanning subgraphs of G, meaning subgraphs that use the same vertex set V but may select any subset E_H ⊆ E of edges. To make the notion concrete, one commonly fixes a weight function w on edges and defines f(H) = sum_{e ∈ E_H} w(e). In this setup spanningvalmaxG(G, f) is the maximum possible sum of weights over all spanning subgraphs of G.

A common specialization fixes the class of spanning subgraphs to spanning trees—subgraphs that are connected and acyclic and contain all vertices. In this case spanningvalmaxG(G, f) equals the weight of a maximum spanning tree with respect to the edge weights w. This value is computable in polynomial time by standard algorithms such as Kruskal’s or Prim’s algorithm.

If one allows arbitrary spanning subgraphs with cycles and uses the edge-weight sum as f, the maximum is achieved by including all edges with positive weight, making the problem effectively trivial in this setting. For more complex or non-additive valuation functions, the problem can become NP-hard depending on the constraints and the exact definition of f.

Applications of spanningvalmaxG appear in network design, resource allocation, and reliability analysis, where one seeks to optimize a cost or value function under the requirement that all vertices remain represented. Related concepts include maximum spanning trees and various theories of spanning subgraphs, each focusing on different admissible subgraph families and valuation schemes.