Split digraphs

We generalize the class of split graphs to the directed case and show that these split digraphs can be identified from their degree sequences. The first degree sequence characterization is an extension of the concept of splittance to directed graphs, while the second characterization says a digraph is split if and only if its degree sequence satisfies one of the Fulkerson inequalities (which determine when an integer-pair sequence is digraphic) with equality.Comment: 14 pages, 2 figures; Accepted author manuscript (AAM) versio

The principal Erdős–Gallai differences of a degree sequence

The Erdős–Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erdős–Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao\u27s order on degree sequences

On fractional realizations of graph degree sequences

We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope. We describe properties of the polytope vertices and characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of pseudo-split graphs, and we characterize their realizations both in terms of forbidden subgraphs and graph structure.Comment: 18 pages, 4 figure

PEWARNAAN GRAF: POLINOMIAL KROMATIKDAN TEOREMA INVERSI MOBIUS

Misalkan G graf sederhana, dan P(G; k)menyatakan banyaknya cara mewarnai titik-titik diG dengan k warna sedemikian hingga tidak ada duatitik yang berhubungan langsung mendapat warnasama. P(G;k) disebut polinomial kromatik dari G.Untuk graf kincir dan graf terpisah,polinomial kromatiknya bisa ditentukan denganmemeriksa struktur grafnya. Hubungan antara posetdan graf dapat membantu menentukan polinomialkromatik sebuah graf dengan memanfaatkan partisihimpunan titik, latis ikatan dan teorema khususdisebut Teorema Inversi Mobius.Kata kunci: Polinomial kromatik, poset, latisikatan, Inversi Mobius

Characterizations for split graphs and unbalanced split graphs

We introduce a characterization for split graphs by using edge contraction. Then, we use it to prove that any ($2K_{2}$, claw)-free graph with $\alpha(G) \geq 3$ is a split graph. Also, we apply it to characterize any pseudo-split graph. Finally, by using edge contraction again, we characterize unbalanced split graphs which we use to characterize the Nordhaus-Gaddum graphs

A Characterization of the Degree Sequences of 2-Trees

A graph G is a 2-tree if G=K_3, or G has a vertex v of degree 2, whose neighbours are adjacent, and G\v{i}s a 2-tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for recognizing and realizing degree sequences of 2-trees.Comment: 17 pages, 5 figure