Elsevier

Discrete Applied Mathematics

Volume 321, 15 November 2022, Pages 10-18
Discrete Applied Mathematics

On forcibly k-connected and forcibly k-arc-connected digraphic sequences

https://doi.org/10.1016/j.dam.2022.06.029Get rights and content

Abstract

For nonnegative integer sequences d={(d1+,d2+,,dn+),(d1,d2,,dn)} and d={(d1+,d2+,,dn+),(d1,d2,,dn)}, we say that d majorizes d, denoted by dd, if dj+dj+ and djdj for 1jn. A digraphic sequence d={(d1+,d2+,,dn+),(d1,d2,,dn)} is forcibly k-connected (resp., k-arc-connected) if each digraph with degree sequence d is k-connected (resp., k-arc-connected). We give a sufficient condition for forcibly k-connected (resp., k-arc-connected) digraphic sequences and observe that if a digraphic sequence d satisfies the condition (which implies d is forcibly k-connected (resp., k-arc-connected)) then every digraphic sequence dd also satisfies the condition. If d violates the condition, then we may take a sequence dd such that there exists a non-k-connected (resp., non-k-arc-connected) d-realization.

Section snippets

Introduction and preliminaries

In this paper, we work on the study of digraphic sequences. For a digraph D=(V,A) with vertex set V=V(D) and arc set A=A(D) and an arc uvA(D), we say that u is the tail and v is the head of the arc uv. Here, we study simple digraphs which have neither multiple arcs (arcs with the same head and the same tail) nor loops. For vV(D), the out-neighborhood of v is ND+(v)={u:vuA(D)} and the in-neighborhood of v is ND(v)={u:uvA(D)}. Their cardinalities are the out-degree of v, dD+(v)=|ND+(v)|

k-connected digraphs

In this section, let d={(d1+,d2+,,dn+),(d1,d2,,dn)} be a digraphic sequence, we will develop the theory of forcibly k-connected digraphic sequences for k1. A digraph is 1-connected (or 1-arc-connected) is equivalent to saying that it is strong.

Theorem 2

If a digraphic sequence d={(d1+,d2+,,dn+),(d1,d2,,dn)} satisfies each of the following for 2in2, then d is forcibly strong:

  • (A1)

    d1+,d11,

  • (A2)

    di+i or dnini,

  • (A3)

    dii or dni+ni.

Proof

Under the conditions stated in Theorem 2, we conversely suppose that d

k-arc-connected digraphs

In this section, let n,a,b1 be integers and let σ+={(x1+,,xa+),(y1,,yb)} and σ={(x1,,xa),(y1+,,yb+)} be sequences of nonnegative integers, where a+bn. Define ps+=|{j{1,,a}:xj+s}|,qt=|{j{1,,b}:yjt}|,ps=|{j{1,,a}:xjs}|,qt+=|{j{1,,b}:yj+t}| for each integer s0 and each integer t0. The notations mentioned above will be used throughout this section. Also we use the same notation for d as in the preceding section. Then we obtain the following conclusions which are of

Acknowledgments

The authors wish to acknowledge anonymous reviewers for their valuable comments.

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Cited by (2)

The research is supported by NSFXJ (2020D04046), NSFC, China (11861066), NSFXJ (2021D01C116).

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