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\begin{frontmatter}

% Title, authors and addresses

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\title{Mixed connectivity of Cartesian graph products and bundles}%\thanksref{label0}

%\thanks[label0]{This 
%work was supported in part by the Slovenian research agency, grants L2-7207-0101, and P1-0294-0101.}
%\thanks[label6]{On leave from FME, University of Maribor.}


% use optional labels to link authors explicitly to addresses:
% \author[label1,label2]{}
% \address[label1]{}
% \address[label2]{}

\author[label2,label3]{Rija Erve\v s}, 
\ead{rija.erves@uni-mb.si}
\author[label2,label4]{Janez \v{Z}erovnik}
\ead{janez.zerovnik@imfm.uni-lj.si}

\address[label2]{
Institute of Mathematics, Physics and Mechanics,\\ Jadranska 19, Ljubljana 1000, Slovenia.\\}
\address[label3]{
FCE,
University of Maribor,\\ Smetanova 17, Maribor 2000, Slovenia.\\}
\address[label4]{
FME,
University of Ljubljana,\\ A\v{s}ker\v{c}eva 6, Ljubljana 1000, Slovenia.\\}

\begin{keyword}
vertex connectivity, edge connectivity, mixed connectivity, Cartesian graph bundle, 
Cartesian graph product, interconnection network, fault tolerance.
% keywords here, in the form: keyword \sep keyword
\end{keyword}

\end{frontmatter}

% main text

\newtheorem{theorem}{Theorem}[section]
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\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{question}[theorem]{Question}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}
% The Appendices part is started with the command \appendix;
% appendix sections are then done as normal sections
% \appendix

% \section{}
% \label{}


\section{Introduction}


%In the design of large interconnection networks several factors have to be taken into account.
%A usual constraint is that each  processor can be connected to a limited number of other processors and 
%the delays in communication must not be too long. Furthermore, 
An interconnection network should be fault tolerant, because practical communication networks are exposed 
to failures of network components. Both failures of nodes and failures of connections between them happen and it is desirable that a network is robust in 
the sense that a limited number of failures does not break down the whole system.
A lot of work has been done on various aspects of network fault tolerance,
see for example the survey \cite{Bermond} and more recent papers \cite{hung,sun,yin}.
In particular the fault diameter with faulty vertices which was first studied in \cite{1} and the edge fault diameter has  
been determined for many important networks recently \cite{zer-ban,zbedge,ZB,erves,day,4,7,0}. 
%In particular, the (vertex) fault diameter and the edge fault diameter of Cartesian graph products and Cartesian graph bundles 
%was studied recently \cite{zer-ban,zbedge,ZB,erves}.
Usually either only edge faults or only vertex faults are considered, 
while the case when both edges and vertices may be faulty is studied rarely. 
%\cite{hung,sun} consider Hamiltonian properties assuming a combination of vertex and edge faults. 
In recent work on fault diameter of Cartesian graph products and bundles \cite{zer-ban,zbedge,ZB,erves}, 
analogous results were found for both fault diameter and edge fault diameter. However, the proofs
for vertex and edge faults 
%in \cite{zer-ban,zbedge,ZB,erves} 
are independent, 
and our effort to see how results in one case may imply the others was  not successful.
A natural question is whether it is possible to design  a uniform theory that would enable unified proofs or 
provide tools to translate results for one type of faults to the other. 
It is therefore of interest to study general relationships between invariants under simultaneous vertex and edge faults.
Some basic results on edge, vertex and mixed fault diameters for general graphs appear in \cite{3fd}.
In order to study the fault diameters of graph products and bundles  under mixed faults, it is important to 
understand the generalized connectivities.
%
We define mixed connectivity which generalizes both vertex and edge connectivity, and observe some basic facts for any connected graph. Furthermore, we generalize results of vertex connectivity and  edge connectivity of Cartesian graph bundles \cite{zer-ban,erves}.
As a corollary,  mixed connectivity of the Cartesian product of finite number of factors is given.
In particular Theorem \ref{mpsv} improves the result on edge connectivity of Cartesian graph products and bundles.  

\section{Mixed connectivity}

%The  {\em connectivity} (or {\em vertex connectivity}) of a connected graph $G$, $\kappa (G)$, is the minimum cardinality over all 
%vertex-separating sets in $G$. 
%As the complete graph $K_n$ has no vertex-separating sets, we define $\kappa (K_n)=n-1$. 
%We say that $G$ is  {\em $k$-connected} (or {\em $k$-vertex connected}) for any $k \leq \kappa (G)$.
%%
%The  {\em edge connectivity} of a connected graph $G$, $\lambda (G)$, is the minimum cardinality over all edge-separating sets in $G$.
%A graph $G$ is said to be {\em $k$-edge connected}  for any $k \leq \lambda (G)$.
%%
%In other words, the edge connectivity $\lambda (G)$ of a connected graph $G$ is the smallest number of edges whose removal disconnects $G$, 
%and the (vertex) connectivity $\kappa (G)$ of a connected graph $G$ (other than a complete graph) is the smallest number of vertices whose 
%removal disconnects $G$. 
%%
%It is well-known that
%%(see, for example, \cite{m}, page 224)
%for any graph $G$
%$\kappa (G) \leq \lambda (G) \leq \delta_G, $
%where $\delta_G$ is the smallest vertex degree of $G$. 
%Thus if a graph $G$ is $k$-connected, then it is also $k$-edge connected. The reverse does not hold in general.


{\begin{definition}
Let $G$ be any connected graph. A graph $G$ is
\emph{$(p,q)$+connected}, if $G$ remains connected after removal of any $p$ vertices and any $q$ edges. 
\end{definition}} 

%Mixed connectivity is a generalization of vertex and edge connectivity.
Any connected graph $G$ is $(0,0)$+connected, 
$(p,0)$+connected for any $p<\kappa(G)$ and $(0,q)$+connected for any $q<\lambda(G)$,
where 	 $\kappa(G)$ and $\lambda(G)$ are the usual vertex- and edge-connectivities.
In our notation $(i,0)$+connected is the same as $(i+1)$-connected, i.e. the graph remains connected 
after removal of any $i$ vertices.
Similarly, $(0,j)$+connected is the same as $(j+1)$-edge connected, i.e. the graph remains connected 
after removal of any $j$ edges.
%
Clearly, if $G$ is $(p,q)$+connected graph, then $G$ is $(p^{\prime},q^{\prime})$+connected for any
$p^{\prime}\leq p$ and any $q^{\prime}\leq q$.
Furthermore, for any connected graph $G$ with $k<\kappa(G)$ faulty vertices, at least $k$ edges are not in functional.
Roughly speaking, graph $G$ remains connected if any faulty vertex in $G$ is replaced with any edge. It is easy to prove that if a graph $G$ is $(p,q)$+connected and $p>0$,  then $G$ is $(p-1,q+1)$+connected.
Hence for $p>0$ we have a chain of implications: 
$(p,q)$+connected $\Longrightarrow$ $(p-1,q+1)$+connected $\Longrightarrow\dots\Longrightarrow$ 
$(1,p-1+q)$+connected $\Longrightarrow$  $(0,p+q)$+connected, 
that %for $q=0$ 
generalizes the  well-known proposition 
that any $k$-connected graph is also $k$-edge connected. 
%
Therefore, a graph $G$ is $(p,q)$+connected if and only if $p<\kappa(G)$ and $p+q<\lambda(G)$. 
%and $G$ is $(p,0)$+connected $\Longleftrightarrow$ $p$+connected $\Longleftrightarrow$ $(p+1)$-connected, 
%$G$ is $(0,q)$+connected $\Longleftrightarrow$ $q$+edge connected $\Longleftrightarrow$  $(q+1)$-edge connected.


%%%%%

If for a graph $G$ $\kappa(G)=\lambda(G) =k$, then $G$ is $(i,j)$+connected exactly when $i+j< k$.
%
However, if  $2\leq \kappa(G)<\lambda(G)$, the question whether $G$ is $(i,j)$+connected  for  $1\leq i<\kappa(G)\leq i+j < \lambda(G)$
is not trivial. The example below shows that in general knowing 
$\kappa(G)$ and $\lambda(G)$ is not enough to decide whether  $G$ is  $(i,j)$+connected.

\begin{example} \label{pr}
For graphs on Fig. \ref{sl} we have $\kappa(G_1)=\kappa(G_2)=2$ and $\lambda(G_1)=\lambda(G_2)=3$. Both graphs are 
$(1,0)+$connected $\Longrightarrow$ $(0,1)+$connected, and $(0,2)+$ connected.
Graph $G_1$ is not $(1,1)+$connected, while graph $G_2$ is.
\end{example}

\begin{figure}[htb] 
\begin{center}
    \includegraphics[width=4.0in]{grafMP12.eps}
    \caption{Graphs $G_1$ and $G_2$ from Example \ref{pr}.}
   \label{sl}
  \end{center}
\end{figure}



%
%
%\begin{figure} 
%\begin{scriptsize}
%$$ \begin{array}{ccccccccccc}
%
%     &   &      &      &      &       &       &            &  (0,\lambda(G)-1){\textrm +C}   \\  
%     &   &      &      &      &       &       &            &  \Downarrow \\ 
%    &   &      &      &      &       &       &						 &  (0,\lambda(G)-2){\textrm +C} \\ 
%    &   &      &      &      &       &       & 					 &  \Downarrow \\ 
%     &	  &      &      &      &       &       & 					 &  \vdots \\
%    &   &      &      &      &       &       & 					 &  \Downarrow \\ 
%% (\kappa\kappa-1,1){\textrm -C} & \Leftrightarrow  &  (\kappa-2,2){\textrm -C}    & \Rightarrow &  \dots  & \Rightarrow  & (1,\kappa-1){\textrm -C}    & \Rightarrow  &  (0,\kappa){\textrm -C} \\  
%%  \Downarrow &   & \Downarrow &     &     &     & \Downarrow &      &  \Downarrow \\ 
%      (\kappa(G)-1,0){\textrm +C}  &  \Rightarrow &  (\kappa(G)-2,1){\textrm +C} & \Rightarrow      & \dots & \Rightarrow  & (1,\kappa(G)-2){\textrm +C}    & \Rightarrow  &  (0,\kappa(G)-1){\textrm +C} \\  
%       &   &\Downarrow &     & &            & \Downarrow &              &  \Downarrow \\ 
%       &   &  (\kappa(G)-2,0){\textrm +C}  &   \Rightarrow  & \dots & \Rightarrow  & (1,\kappa(G)-3){\textrm +C}    & \Rightarrow  &  (0,\kappa(G)-2){\textrm +C} \\  
%    &        &  &  &  &    &  \Downarrow &       &  \Downarrow \\ 
%     &       & & 							  &   &	 & \vdots &             & \vdots \\
%     &      &  &  &  &   &  \Downarrow &       &  \Downarrow \\ 
%     &      &  &  &  &  & (1,0){\textrm +C}    & \Rightarrow  &  (0,1){\textrm +C} \\  
%     &     & &   &       &       &  &              &  \Downarrow \\ 
%    &     & &   &       &       &  &   &  (0,0){\textrm +C} \\ 
%
%\end{array} 
%$$
%\end{scriptsize}
%\caption{Diagram 1} \label{Diagram1}
%\end{figure}

%Let $G$ be any connected graph, $ \kappa(G)<\lambda(G)$. 
%All known mixed connectivities for a graph $G$ are summarized in Diagram 1 (Fig. \ref{Diagram1}).
%
Both edge connectivity   and  vertex connectivity of a  graph can be computed in 
polyonimal time.  
%More precisely, according to \cite{KorteVygen}, there is an algorithm with time complexity $O(n^4)$ for vertex connectivity \cite{Henzinger}
%and edge connectivity can be computed within $O(m+\lambda(G) n \log\frac{n}{\lambda(G)})$ time \cite{Gabow}.
%Hence given a graph $G$, there are polynomial time algorithms to decide $(i,0)$ and $(0,j)$+connectivities for all $i$, $j$.
Therefore it is interesting to ask 

\noindent{\bf Problem.}
Let $G$ be a graph and $1\leq i<\kappa(G)\leq i+j < \lambda(G)$.
Is there a polynomial algorithm to decide whether $G$ is $(i,j)$+connected?


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%If a graph $G$ is $(i,j)$-connected for some $1\leq i<\kappa(G) \leq i+j < \lambda(G)$ then the upper part of 
%Diagram 1 (Fig. \ref{Diagram1}) can be updated with known mixed connectivities.  
%%
%Two extreme cases are:
%\hfill\break\noindent
%(1)
%If graph $G$ is
%$(\kappa(G)-1,\lambda(G) - \kappa(G))$+connected then the diagram of mixed connectivities is maximal
%because this connectivity implies all possible mixed connectivities for any connected graph. 
%%
%Namely, if a graph $G$ is
%$(\kappa(G)-1,\lambda(G) - \kappa(G))$+connected then graph $G$ is $(i,j)$+connected 
%for all $i<\kappa(G)$ and all $i+j< \lambda(G)$.
%\hfill\break\noindent
%(2)
%If $G$ is not $(1,\kappa(G)-1)$+connected then the diagram of mixed connectivities is minimal, i.e. the 
%connectivities from Diagram 1 are all connectivities of $G$.
%%
%\newline 
%In general,   $(\kappa(G)-1)\times (\lambda(G) - \kappa(G))$ 
%different connectivities  may have to be checked to complete the diagram of all connectivities of $G$. 

 
\section{Mixed connectivity of Cartesian graph products and bundles}

Graph products and bundles are among frequently  studied interconnection network topologies.
For example the meshes, tori, hypercubes and some of their generalizations are Cartesian products.
It is less known that  some well-known topologies 
are  Cartesian graph bundles, i.e. some twisted hypercubes %\cite{cull,efe} 
and 
multiplicative circulant graphs. % \cite{ivan}.  
%
Graph bundles also appear as computer topologies. 
A well known example is the twisted torus,
a Cartesian graph bundle with fibre $C_4$ over base $C_4$ is the ILLIAC IV architecture, % \cite{ILLIAC},
a famous supercomputer that inspired some modern multicomputer architectures.
It may be interesting to note that the original design was a graph bundle with fibre $C_8$ over base $C_8$, 
but due to high cost a smaller version was build. % \cite{computermuseum}.
%
A Cartesian graph bundle is a generalization of graph cover and the Cartesian graph product.
%They were first studied in \cite{PiVr,ST_P_V}. 


{\begin{definition}
Let $B$ and $F$ be graphs. A graph $G$ is a {\em Cartesian graph bundle with fibre $F$ over the base graph} $B$ if 
there is a {\em graph map} $p:G\rightarrow B$ such that for each vertex $v\in V(B)$, $p^{-1}(\{v\})$ is isomorphic to $F$, 
and for each edge $e=uv\in E(B)$, $p^{-1}(\{e\})$ is isomorphic to $F \Box K_2$.
\end{definition}} 
%
%More precisely, the mapping $p:G\rightarrow B$  maps graph elements of $G$ to graph elements of $B$, 
%i.e. $p: V(G) \cup E(G) \rightarrow  V(B) \cup E(B)$. In particular, here we also assume that 
%the vertices of $G$ are mapped to vertices of $B$ and the edges of $G$ are mapped either to vertices or to edges of $B$. 
%%We say an edge $e\in E(G)$ is {\em degenerate} if $p(e)$ is a vertex. Otherwise we call it {\em nondegenerate}.
%%The mapping $p$ will also be called the {\em projection} (of the bundle $G$ to its base $B$).
%Note that  each edge $e=uv \in E(B)$ naturally induces an isomorphism $\varphi_e :p^{-1}(\{u\})\rightarrow p^{-1}(\{v\})$
%between two fibres. 
%Informally, one can say that bundles are "twisted products".

%In previous  work 
%on vertex and edge fault diameters of Cartesian graph bundles it is proved
%that Cartesian graph bundle  with fibre $F$ over the base graph $B$ is  
%$(\kappa(F)+\kappa(B))$-connected and $(\lambda(F)+\lambda(B))$-edge connected \cite{zer-ban,erves}.
%Roughly speaking, 
%consider the maximum allowed number of faulty elements of graphs $F$, $B$ and $G$. 
%In both cases is assured connectivity of a graph $G$ with one more faulty element, 
%vertex or edge respectively, as the sum of faulty vertices and faulty edges in graphs $F$ and $B$. 
%
%We proved that besides the sum of allowed  faulty elements of the fibre and the base, 
%one additional faulty element  is allowed, and whenever is applicable, the extra faulty element can be a node.
%%
%In particular this improves the result on edge connectivity.  
%Namely, Theorem \ref{mpsv} is for $p_F=p_B=0$ stronger than proposition about edge connectivity of Cartesian graph bundles. 
%If graph $F$ is $(0,q_{F}-1)$+connected and graph $B$ is $(0,q_{B}-1)$+con\-nected, then 
%by Theorem \ref{mpsv}, Cartesian graph bundle with fibre $F$ over the base graph $B$ is 
%$(1,q_{F}+q_{B})$+connected,
%while edge connectivity assures only 
%$(0,q_{F}+q_{B}-1)$+connectivity.

We have generalized the result \cite{zer-ban} on (vertex) connectivity 
and improved the result \cite{erves} on edge connectivity:

\begin{theorem}
\label{mpsv} Let $G$ be a Cartesian graph bundle with fibre $F$ over the base graph $B$, graph $F$ be $(p_{F},q_{F})$+connected and
graph $B$ be $(p_{B},q_{B})$+connected.
Then Cartesian graph bundle $G$ is $(p_{F}+p_{B}+1,q_{F}+q_{B})$+connected.
\end{theorem}

As the Cartesian product is a Cartesian graph bundle where all the isomorphisms between the fibres are identities, 
the statement about mixed connectivity of Cartesian graph products of a finite number of factors follows 
easily from Theorem \ref{mpsv}. 

\begin{corollary}
\label{mppr} Let graphs $G_i, i=1,\dots,k,$ be $(p_i,q_i)$+connected.
Then the Cartesian graph product $G=G_1\Box G_2\Box \dots \Box G_k$  is $(\sum p_i+k-1,\sum q_i)+$connected.
\end{corollary}


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% 
%\bibitem{computermuseum}
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\end{thebibliography}

\end{document}

