Hereditary unigraphs and Erdős–Gallai equalities

Abstract

We give characterizations of the structure and degree sequences of hereditary unigraphs, those graphs for which every induced subgraph is the unique realization of its degree sequence. The class of hereditary unigraphs properly contains the threshold and matrogenic graphs, and the characterizations presented here naturally generalize those known for these other classes of graphs.The degree sequence characterization of hereditary unigraphs makes use of the list of values k for which the kth Erdős-Gallai inequality holds with equality for a graphic sequence. Using the canonical decomposition of Tyshkevich, we show how this list describes structure common among all realizations of an arbitrary graphic sequence.

Recognition of split-graphic sequences

“…In 1993 Blázsik, Hujter, Pluhár and Tuza [5] characterized the pseudo split graphs defined as graphs with no induced C 4 and 2K 2 (see also [2]). In 1998 Maffray and Preissmann [58] proved the following assertion.…”

“…Clearly, J 1 l,m = J l,m . Figure 2 shows J 3,2 (part a) and J 2 3,2 (part b). The structure of the paper is as follows.…”

“…, y n ) containing the cutting points of the elements of s. For given s i the weight point w i is the largest k (1 ≤ k ≤ n) having the property s k ≥ i. . The cutting point y i belonging to s i is the maximum of i and w i , see (2).…”

Split graphs and Nordhaus–Gaddum graphs

“…Thus, every NG-graph is decomposable except when it is a single vertex or a 5-cycle. Chvátal and Hammer [5], Blázik et al [3] and Barrus [1,2] also characterized various graph classes in terms of their decompositions.…”

“…For an NG-graph G, let B ′ G be the set of vertices in B G that have no neighbors in C G . In the proof of Theorem 30, we obtain our bijections by removing A G in part (1) and A G ∪ B ′ G in part (2). In both cases, the result is a split graph on n − 2 or fewer vertices.…”