Upper bound for local average order at stems

Establish that, for every graph G of order n≥2 without isolated vertices and every l-stem v of G with l≥1, the local average order of dominating sets containing v satisfies avd_v(G)≤(4n+1)/6, with equality if and only if l=1 and G is star-like.

Background

The paper studies the local average order of dominating sets containing a specified vertex, denoted avd_v(G). It proves several upper bounds for this quantity, including avd_v(G)<(4n+1)/6 for every l-stem with l≥2. It also proves the bound avd_v(G)≤(4n-l+2)/6 for stems whose neighbors are all leaves or stems, with a characterization of equality in that restricted setting.

The unresolved case is principally that of 1-stems having non-leaf neighbors that are not stems. The conjecture proposes that the universal upper bound (4n+1)/6 nevertheless holds for every stem, and that equality occurs exactly when the stem has one leaf and the entire graph is star-like.