For any complete chain I whose distinct elements are separated by cover pairs, and for every family {Mi | i∈ I} of monoids, we construct a family of graphs {Gi | i∈ I} such that Gi is a proper induced subgraph of Gj for all i; j∈ I with i ¡ j, Gi = ⋂ {Gj | j∈ I; i ¡ j} whenever i = inf{ j∈ I | j¿ i} in I , Gi = ⋃ {Gj | j∈ I; j ¡ i} whenever i = sup{ j∈ I | j¡ i} in I , the endomorphism monoid o...

An early characterization of distance-hereditary graphs is that every cycle of length 5 or more has crossing chords. A new, stronger, property is that in every cycle of length 5 or more, some chord has at least two crossing chords. This new property can be characterized by every block being complete multipartite, and also by the vertex sets of cycles of length 5 or more always inducing 3-connec...

Let C be any class of graphs. A graph G is C-ultrahomogeneous if every isomorphism between induced subgraphs belonging to C extends to an automorphism of G. We study graphs that are K∗-ultrahomogeneous, where K∗ is the class of complete graphs. We also explicitly classify the graphs that are tK∗-ultrahomogeneous, where tK∗ is the class of disjoint unions of complete graphs.