Alain BrettoMathematical EngineeringHypergraph Theory2013An Introduction10.1007/978-3-319-00080-0_1© Springer International Publishing Switzerland 2013

1. Hypergraphs: Basic Concepts

Alain Bretto¹  

(1)

Computer Science Department, Universite de Caen, GREYC-CNRS UMR 6072, 14032 Caen, France

Alain Bretto

Email: alain.bretto@unicaen.fr

Abstract

View the significant developments of combinatoric thanks to computer science [And89, LW01], hypergraphs are increasingly important in science and engineering. Hypergraphs are a generalization of graphs, hence many of the definitions of graphs carry verbatim to hypergraphs. In this chapter we introduce basic notions about hypergraphs. Most of the vocabulary used in this book is given here and most of this one is a generalization of graphs languages [LvGCWS12].

View the significant developments of combinatoric thanks to computer science [And89, LW01], hypergraphs are increasingly important in science and engineering. Hypergraphs are a generalization of graphs, hence many of the definitions of graphs carry verbatim to hypergraphs. In this chapter we introduce basic notions about hypergraphs. Most of the vocabulary used in this book is given here and most of this one is a generalization of graphs languages [LvGCWS12].

1.1 First Definitions

Ahypergraph $$H$$ denoted by $$H = (V;\! E=(e_{i})_{i\in I})$$ on a finite set $$V$$ is a family $$(e_i)_{i\in I}$$ , ( $$I$$ is a finite set of indexes) of subsets of $$V$$ calledhyperedges. Sometimes $$V$$ is denoted by $$V(H)$$ and $$E$$ by $$E(H)$$ .

Theorder of the hypergraph $$H= (V;\! E)$$ is the cardinality of $$V$$ , i.e. $$\vert V\vert = n$$ ; itssize is the cardinality of $$E$$ , i.e. $$\vert E\vert =m$$ .

By definition theempty hypergraph is the hypergraph such that:

$$V = \emptyset $$ ;

$$E = \emptyset $$ .

Always by definition atrivial hypergraph is a hypergraph such that:

$$V \ne \emptyset $$ ;

$$E = \emptyset $$ .

In the sequel, unless stated otherwise, hypergraphs have a nonempty set of vertices, a non-empty set of hyperedges and they do not contain empty hyperedge.

Let $$(e_{\!j})_{j\in J}$$ , $$J\subseteq I$$ be a subfamily of hyperedges of $$E =(e_{i})_{i\in I}$$ , we denote the set of vertices belonging to $$\cup _{j\in J}e_{j}$$ by $$V(\cup _{j\in J}e_{j})$$ , but sometimes we use $$e$$ for $$V(e)$$ . For instance, sometimes we use $$e\cap V^{\prime }$$ for $$V(e)\cap V^{\prime }$$ , $$V^{\prime }\subseteq V$$ .

If

the hypergraph is withoutisolated vertex, where a vertex $$x$$ is isolated if

A hyperedge $$e\in E$$ such that $$\vert e\vert =1$$ is aloop.

Two vertices in a hypergraph areadjacent if there is a hyperedge which contains both vertices. In particular, if $$\{x\}$$ is an hyperedge then $$x$$ is adjacent to itself. Two hyperedges in a hypergraph areincident if their intersection is not empty.

Let $$H= (V;\! E =(e_{i})_{i\in I})$$ be a hypergraph:

Theinduced subhypergraph $$H(V^{\prime })$$ of the hypergraph $$H$$ where $$V^{\prime }\subseteq V$$ is the hypergraph $$H(V^{\prime })=(V^{\prime }, E^{\prime })$$ defined as

The letter $$E^{\prime }$$ can be represented a multi-set. Moreover, according to the remark above we can add, if we need, the emptyset.

Given a subset $$V^{\prime }\subseteq V$$ , thesubhypergraph $$H^{\prime }$$ is the hypergraph

Apartial hypergraph generated by $$J\subseteq I$$ , $$H^{\prime }$$ of $$H$$ is a hypergraph

where $$\bigcup _{j\in J}e_{j}\subseteq V^{\prime }$$ . Notice that we may have $$V^{\prime }=V$$ .

The star $$H(x)$$ centered in $$x$$ is the family of hyperedges $$(e_j)_{j\in J}$$ containing $$x$$ ; $$d(x) = \vert J\vert $$ is thedegree of $$x$$ excepted for a loop $$\{x\}$$ where the degree $$d(x)=2$$ . If the hypergraph is without repeated hyperedge the degree is denoted by $$d(x) = \vert H(x)\vert $$ , excepted for a loop $$\{x\}$$ where the degree $$d(x)=2$$ . The maximal degree of a hypergraph $$H$$ is denoted by $$\varDelta (H)$$ .

If each vertex has the same degree, we say that the hypergraph isregular, or $$k$$ -regular if for every $$x\in V$$ , $$d(x) = k$$ .

If the family of hyperedges is a set, i.e. if $$i\ne j \Longleftrightarrow e_{i}\ne e_{j}$$ , we say that $$H$$ is without repeated hyperedge. Therank $$r(H)$$ of $$H$$ is the maximum cardinality of a hyperedge in the hypergraph: $$r(H) = \max _{i\in I}\vert e_{i}\vert $$ ; the minimum cardinality of a hyperedge is theco-rank $$cr(H)$$ of $$H$$ : $$cr(H) = \min _{i\in I}\vert e_{i}\vert $$ . If $$r(H)=cr(H) = k$$ the hypergraph isk-uniform or uniform.

1.2 Example of Hypergraph

Let $$M$$ be a computer science meeting with $$k\ge 1$$ sessions : $$S_{1},S_{2},S_{3},\ldots , S_{k}$$ . Let $$V$$ be the set of people at this meeting. Assume that each session is attended by one person at least. We can build a hypergraph in the following way:

The set of vertices is the set of people who attend the meeting;

the family of hyperedges $$(e_{i})_{i\in \{1, 2, \ldots k\}}$$ is built in the following way:

$$e_{i}$$ , $$i\in \{1, 2, \ldots k\}$$ is the subset of people who attend the meeting $$S_{i}$$

Fano plane. TheFano plane is the finite projective plane of order $$2$$ , which have the smallest possible number of points and lines, $$7$$ points with $$3$$ points on every line and $$3$$ lines through every point. To a Fano plane we can associate a hypergraph called Fano hypergraph:

The set of vertices is $$V=\{0,1,2,3,4,5,6\}$$ ;

The set of hyperedges is $$E=\{013, 045, 026, 124, 346, 235, 156\}$$

The rank is equal to the co-rank which is equal to $$3$$ , hence, Fano hypergraph is $$3$$ -unifirm. Figure 1.1 show Fano hypergraph

Steiner systems. Let $$t; k; n$$ be integers which satisfied: $$2 \le t \le k < n$$ . ASteiner system denoted by $$S(t; k; n)$$ is a $$k$$ -uniform hypergraph $$H = (V;\! E)$$ with $$n$$ vertices such that for each subset $$T\subseteq V$$ with $$t$$ elements there is exactly one hyperedge $$e\in E$$ satisfying $$T\subseteq e$$ . For instance the complete graph $$K_n$$ is a $$S(2; 2; n)$$ Steiner system. An important example is the Steiner systems $$S(2; 3; n)$$ which are calledSteiner triple systems. The Fano plane is an example of a Steiner triple system on 7 vertices.

Linear spaces. A linear space is a hypergraphs in which each pair of distinct vertices is contained in precisely one edge. To exclude trivial cases, it is always assumed that there are no empty or singleton edges.

A hypergraph with only one edge which contains all vertices, this is called atrivial linear space.

Asimple hypergraph is a hypergraph $$H = (V; E)$$ such that: $$e_{i} \subseteq e_{j}\Longrightarrow i = j$$ . A simple hypergraph has no repeated hyperedge.

A hypergraph islinear if it is simple and $$\vert e_{i}\cap e_{j}\vert \le 1$$ for all $$i, j\in I$$ where $$i \ne j$$ .

The following algorithm gives thesimple hypergraph associated to a hypergraph (Fig. 1.2).

Fig. 1.1

The hypergraph above is Fano hypergraph

1.2.1 Simple Reduction Hypergraph Algorithm

Let $$H = (V; E)$$ be a hypergraph without isolated vertex; apath $$P$$ in $$H$$ from $$x$$ to $$y$$ , is a vertex-hyperedge alternative sequence:

such that

$$x_1, x_2, \ldots , x_s, x_{s+1}$$ are distinct vertices with the possibility that $$x_{1}= x_{s+1}$$ ;

$$ e_1, e_2,\ldots , e_s$$ are distinct hyperedges;

$$x_i, x_{i+1}\in e_i$$ , $$(i = 1, 2, . . . , s)$$ .

If $$x=x_1=x_{s+1}=y$$ the path is called a cycle.

The integer $$s$$ is the length of path $$P$$ . Notice that if there is a path from $$x$$ to $$y$$ there is also a path from $$y$$ to $$x$$ . In this case we say that $$P$$ connects $$x$$ and $$y$$ . A hypergraph isconnected if for any pair of vertices, there is a path which connects these vertices; it not connected otherwise. In this case we may also say that it isdisconnected.

Thedistance $$d(x, y)$$ between two vertices $$x$$ and $$y$$ is the minimum length of a path which connects $$x$$ and $$y$$ . If there is a pair of vertices $$x, y$$ with no path from $$x$$ to $$y$$ (or from $$y$$ to $$x$$ ), we define $$d(x, y)= \infty $$ ( $$H$$ is not connected). Let $$H = (V,E)$$ be a hypergraph, a connected component is a maximal set of vertices $$X\subseteq V$$ such that, for all $$x, y\in X$$ , $$d(x, y)\ne \infty $$ . Thediameter $$d(H)$$ of $$H$$ is defined by

The relation:

$$x\mathcal{R }y$$ if and only if either there is a path from $$x$$ to $$y$$ , or $$x= y$$ .

is an equivalence relation; the classes of this relation are the connected components of the hypergraph (Fig. 1.3).

Fig. 1.2

The hypergraph $$H$$ above has $$11$$ vertices; $$5$$ hyperedges; $$1$$ loop: $$e_{5}$$ ; $$2$$ isolated vertices: $$x_{11}$$ , $$x_{9}$$ . The rank $$r(H) = 4$$ , the co-rank $$cr(H) = 1$$ . The degree of $$x_{1}$$ is $$2$$ . $$H^{\prime }= (V; \{e_{1}, e_{2}\})$$ is a partial hypergraph generated by $$J = \{1, 2\}$$ ;

is an induced subhypergraph. Notice that $$e_{3}\cap V^{\prime }= \{x_{6}\}$$ is not an hyperedge for this induced hypergraph. $$H^{\prime } = (V^{\prime }= \{x_{1},x_{2}, x_{3}, x_{4}, x_{7} \}, E= \{e_{1}\})$$ is a subhypergraph with $$1$$ isolated vertex: $$ x_{7}$$ . Hypergraph $$H$$ is linear and simple

Fig. 1.3

The hypergraph above has $$2$$ connected components, $$C_{1},C_{2}$$ . $$P =x_{10}e_{4}x_{5}e_{3}x_{6}e_{2}x_{4}e_{1}x_{3}$$ is a path from $$x_{10}$$ to $$x_{3}$$ , $$P^{\prime } =x_{10}e_{4}x_{1}e_{1}x_{3}$$ is also a path from $$x_{10}$$ to $$x_{3}$$ and the distance $$d(x_{10}, x_{3}) = 2$$ is the length of $$P^{\prime }$$ . Notice that the distance $$d(x_{10}, x_{3})$$ is