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Containment Graphs and Posets of Paths in a Tree: Wheels and Partial Wheels

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Abstract

We consider questions regarding the containment graphs of paths in a tree (CPT graphs), a subclass of comparability graphs, and the containment posets of paths in a tree (CPT orders). In 1984, Corneil and Golumbic observed that a graph G may be CPT, yet not every transitive orientation of G necessarily has a CPT representation, illustrating this on the even wheels W2k(k ≥ 3). Motivated by this example, we characterize the partial wheels that are containment graphs of paths in a tree, and give a number of examples and obstructions for this class. Our characterization gives the surprising result that all partial wheels that admit a transitive orientation are CPT graphs. We then characterize the CPT orders whose comparability graph is a partial wheel.

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Authors and Affiliations

  1. Department of Computer Science, University of Haifa, Haifa, Israel

    Martin Charles Golumbic

  2. LIMOS, Université Clermont-Auvergne, Clermont Ferrand, France

    Vincent Limouzy

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  1. Martin Charles Golumbic
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  2. Vincent Limouzy
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Correspondence to Martin Charles Golumbic.

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Golumbic, M.C., Limouzy, V. Containment Graphs and Posets of Paths in a Tree: Wheels and Partial Wheels. Order 38, 37–48 (2021). https://doi.org/10.1007/s11083-020-09526-3

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  • DOI: https://doi.org/10.1007/s11083-020-09526-3

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