Complexity of total matching on chordal graphs

Determine the computational complexity of the total matching problem on chordal graphs.

Background

The paper establishes that the strong total cover problem is polynomial-time solvable on chordal graphs, while noting that the related total matching problem is NP-hard on bipartite graphs. The authors explicitly identify the complexity status of total matching on chordal graphs as unresolved and discuss whether the polynomial-time result for strong total covers could contribute to resolving it.