text:
\section{Introduction and preliminaries}
Throughout this paper, let $G$ be a finite connected graph with vertex set $V=V(G)$, edge set $E=E(G)$, minimum degree $\delta=\delta(G)$ and maximum degree $\Delta=\Delta(G)$. We use \cite{w} as a reference for terminology and notation which are not defined here. For any vertex $v \in V$, $N(v)=\{u\in G\mid uv\in E(G)\}$ denotes the {\em open neighbourhood} of $v$ in $G$, and $N[v]=N(v)\cup \{v\}$ denotes its {\em closed neighbourhood}.

\noindent There are many domination parameters in graph theory. The diversity of domination parameters and the types of proofs involved are very extensive. We believe that some of the results in this field are similar and the main ideas of their proofs are the same. Therefore we introduce and investigate the concept of $(k,k',k'')$-domination number, as a generalization of many domination parameters, by a simple uniform approach.

\noindent Let $k,k'$ and $k''$ be nonnegative integers. A set  $S\subseteq V$ is a {\em $(k,k',k'')$-dominating set} in $G$ if every vertex in $S$ has at least $k$ neighbors in $S$ and every vertex in $V\setminus S$ has at least $k'$ neighbors in $S$ and at least $k''$ neighbors in $V\setminus S$. The {\em $(k,k',k'')$-domination number} $\gamma_{(k,k',k'')}(G)$ is the minimum cardinality of a $(k,k',k'')$-dominating set. We note that every graph with the minimum degree at least $k$ has a $(k,k',k'')$-dominating set, since $S=V(G)$ is such a set.
Note that
\begin{itemize}
\item $\gamma_{(0,1,1)}(G)=\gamma_{r}(G)$: {\em Restrained domination number};
\item $\gamma_{(1,1,1)}(G)= \gamma_{t}^r(G)$: {\em Total restrained domination number};
\item $\gamma_{(1,2,1)}(G)=\gamma_{2r}(G)$: {\em Restrained double domination number};
\item  $\gamma_{(k,k,k)}(G)=\gamma_{\times k,t}^r(G): k$-{\em Tuple total restrained domination number};
\item $\gamma_{(k,k,0)}(G)=\gamma_{\times k,t}(G): k$-{\em Tuple total domination number};
\item $\gamma_{(k-1,k,0)}(G)=\gamma__{\times k}(G): k$-{\em Tuple domination number};
\item $\gamma_{(0,k,0)}(G)=\gamma_{k}(G): k$-{\em Domination number}.
\end{itemize}
For the definitions of the parameters above and a comprehensive work on domination in graphs see \cite{cfhv,cr,dhhm,hhs,hhs2,kn,k}.

\section {Lower bounds on $(k,k',k'')$-domination numbers}

In this section, we calculate a lower bound on $\gamma_{(k,k',k'')}(G)$, which improves
the existing lower bounds on these seven parameters.

\noindent The following result can be found in \cite{cr} and \cite{hjjp2}.
\begin{theorem}\label{LB:TR}
If $G$ is a graph without isolated vertices of order $n$ and size $m$, then
\begin{equation}\label{EQ3}
 \gamma_{t}^r(G)\geq 3n/2-m,
\end{equation}
in addition this bound is sharp.
\end{theorem}

\noindent Also Hattingh et.al \cite{hjlpv} found that
\begin{equation}\label{EQ4}
\gamma_{r}(G)\geq n-2m/3.
\end{equation}

The following known
result is an immediate consequence of Theorem \ref{LB:TR}.
\begin{theorem}\cite{hjjp1}
If $T$ is a tree of order $n\geq2$, then
\begin{equation}\label{EQ5}
 \gamma_{t}^r(T)\geq \lceil\frac{n+2}{2}\rceil.
 \end{equation}
\end{theorem}

The inequality
\begin{equation}\label{EQ6}
\gamma_{r}(T)\geq \lceil\frac{n+2}{3}\rceil
\end{equation}
on restrained domination number of a tree of order $n\geq1$
was obtained by Domke et al. \cite{dhhm}.

\noindent The author in \cite{k} generalized Theorem \ref{LB:TR} and proved
that if $\delta(G)\geq k$, then
\begin{equation}\label{EQ7}
\gamma_{\times k,t}^r(G)\geq 3n/2-m/k.
\end{equation}
Moreover the authors in \cite{kn} proved that if $G$ is a graph without isolated vertices, then
\begin{equation}\label{EQ8}
\gamma_{2r}(G)\geq \frac{5n-2m}{4}.
\end{equation}

\noindent We now improve the lower bounds given in
$(\ref{EQ3}), (\ref{EQ4}), \ldots,(\ref{EQ8})$.
For this purpose we first introduce a notation.
Let $G$ be a graph with $\delta(G)\geq k$ and let $S$ be a $(k,k',k'')$-dominating set in $G$. We define
$$\delta^{*}=\min\{\deg(v)\mid v\in V(G) \mbox{ and } \deg(v) \geq k'+k'' \}.$$

\noindent It is easy to see that $\deg(v)$ is at least $k'+k''$ and
therefore is at least $\delta^{*}$ for all vertices in $V\setminus S$.

\begin{theorem}\label{Th:Lower.Bound}
Let $G$ be a graph with $\delta(G)\geq k$. Then
$$\gamma_{(k,k',k'')}(G)\geq \frac{(k'+\delta^{*})n-2m}{\delta^{*}+k'-k}.$$
\end{theorem}

\begin{proof}
Let $S$ be a minimum $(k,k',k'')$-dominating set in $G$. Then,
every vertex $v\in S$ is adjacent to at least $k$ vertices in $S$.
Therefore $|E(G[S])|\geq k|S|/2$. Let $E(v)$ be the set of edges at
vertex $v$. Now let $v\in V\setminus S$. Since $S$ is a
$(k,k',k'')$-dominating set, it follows that $v$ is incident to at least $k'$ edges
$e_{1}, \ldots ,e_{k'}$ in $[S,V\setminus S]$ and at least $k''$
edges $e_{k'+1}, \ldots ,e_{k'+k''}$ in $E(G[V\setminus S])$.
Since $deg(v)\geq \delta^{*}\geq k'+k''$, $v$ is incident to at least
$\delta^{*}-k'-k''$ edges in $E(v)\setminus \{e_{i}\}_{i=1}^{k'+k''}$.
The value of $|[S,V\setminus S]|+|E(G[V\setminus S])|$ is minimized if
the edges in $E(v)\setminus \{e_{i}\}_{i=1}^{k'+k''}$
belong to $E(G[V\setminus S])$. Therefore
$$\begin{array}{lcl}
2m&=&2|E(G[S])|+2|[S,V\setminus S]|+2|E(G[V\setminus S])|\\
&\geq &k|S|+2k'(n-|S|)+k''(n-|S|)+(\delta^{*}-k'-k'')(n-|S|).
\end{array}$$
This leads to $\gamma_{(k,k',k'')}(G)=|S|\geq \frac{(k'+\delta^{*})n-2m}{\delta^{*}+k'-k}$.
\end{proof}

\noindent We note that when $(k,k',k'')=(1,1,1)$, then Theorem \ref{Th:Lower.Bound} gives
improvements for inequalities (\ref{EQ3}) and (\ref{EQ5}). When $(k,k',k'')=(0,1,1)$,
then it will be improvements of its corresponding results given by (\ref{EQ4}) and (\ref{EQ6}).
Also, if $(k,k',k'')=(k,k,k)$, Theorem \ref{Th:Lower.Bound} improves (\ref{EQ7}) and
if $(k,k',k'')=(1,2,1)$, it improves (\ref{EQ8}).

As an immediate result of Theorem \ref{Th:Lower.Bound}, we conclude the following result of Hattingh and Joubert.

\begin{corollary}\cite{hj}
If $G$ is a cubic graph of order $n$, then $\gamma_{r}(G)\geq \frac{n}{4}$.
\end{corollary}
Also, for total restrained and restrained double domination numbers of a cubic graph $G$, we obtain
$\gamma_{t}^r(G)\geq \frac{n}{3}$ and $\gamma_{2r}(G)\geq \frac{n}{2}$,
by Theorem \ref{Th:Lower.Bound}, respectively.

\noindent Since $\gamma_{\timesk}(G)=\gamma_{(k-1,k,0)}(G)$, Theorem \ref{Th:Lower.Bound} is improvements of the following results.
\begin{theorem}\cite{hh}
Let $G$ be a graph of order $n$ and $\delta(G)\geq k-1$, then
$$\gamma_{\timesk}(G)\geq \frac{2kn-2m}{k+1}$$
and this bound is sharp.
\end{theorem}

\begin{theorem}\cite{zwx}
Let $G$ be a graph of order $n$ and size $m$ with minimum degree $\delta\geq k$. Then $\gamma_{\timesk,t}(G)\geq2(n-\frac{m}{k})$ and this bound is sharp.
\end{theorem}
Theorem \ref{Th:Lower.Bound} is also an improvement of the following Theorem.

\begin{theorem}\cite{fj2}
If $G$ is a graph with $n$ vertices and $m$ edges, then $\gamma_{k}(G)\geq n-\frac{m}{k}$ for each $k\geq1$.
\end{theorem}

\noindent We note that every graph $G$ with $\delta(G)\geq k$ has
a $(k,k',0)$-dominating set such as $S=V(G)$
and therefore $\gamma_{(k,k',0)}(G)$ is well-defined when $\delta(G)\geq k$.

\begin{theorem}\label{Th:LB2}
If $G$ is a graph of order $n$ and $\delta(G)\geq k$, then $\gamma_{(k,k',0)}(G)\geq k'n/(\Delta+k'-k)$.
\end{theorem}

\begin{proof}
Let $S$ be a minimum $(k,k',0)$-dominating set in $G$. Then each vertex of $S$ is adjacent to at least $k$ vertices in $S$ and therefore to at most $\Delta-k$ vertices in $V\setminus S$, and so $|[S,V\setminus S]|\leq (\Delta-k)|S|$. On the other hand, every vertex of $V\setminus S$ has at least $k'$ neighbors in $S$, and so $k'(n-|S|)\leq |[S,V\setminus S]|$. Consequently, $\gamma_{(k,k',0)}(G)=|S|\geq k'n/(\Delta+k'-k)$.
\end{proof}

\noindent The following corollaries are immediate results of Theorem \ref{Th:LB2}.

\begin{corollary}(\cite{hk,zwx})
If $G$ is a graph of minimum degree at least $k$, then $\gamma_{\times k,t}(G)\geq kn/\Delta$ and this bound is sharp.
\end{corollary}

\begin{corollary}\cite{hh}
If $G$ is a graph of order $n$ with $\delta(G)\geq k-1$, then $\gamma_{\times k}(G)\geq kn/(\Delta+1)$ and this bound is sharp.
\end{corollary}

\begin{corollary}\cite{fj1}
If $G$ is a graph of order $n$, then $\gamma_{k}(G)\geq kn/(\Delta+k)$ for every integer $k$.
\end{corollary}

\section {Upper bounds on $(k,k',1)$-domination numbers}

\noindent In this section we present an upper bound on $(k,k',1)$-domination numbers and list some of the existing upper bounds which can be derived from this upper bound.

\begin{theorem}
Let $G$ be a graph of order $n$ and let $k$ and $k'$ be positive integers.
\begin{enumerate}
\item  If $k'\geq k+1$ and $\delta\geq k'+1$, then
$\gamma_{(k,k',1)}(G)\leq n-\delta+ k'-1$.

\item If $k\geq k'$ and $\delta\geq k+2$, then
$\gamma_{(k,k',1)}(G)\leq n-\delta+ k$.
\end{enumerate}
The bounds given in Parts 1 and 2 are sharp.
\end{theorem}

\begin{proof}
Let $u$ be a vertex in $G$ with $\deg(u)=\delta$.

\noindent {\bf Proof of 1:} Suppose that $v_{1},\ldots,v_{k'}\in N(u)$. Since $\delta\geq k'+1$, it follows that $|N[u]\setminus \{v_{1},\ldots,v_{k'}\}|\geq2$ and therefore $N[u]\setminus \{v_{1},\ldots,v_{k'}\}$ is a nonempty set. Also, it is easy to see that the subgraph induced by $N[u]\setminus \{v_{1},\ldots,v_{k'}\}$ has no isolated vertices. Now let $S=V(G)\setminus (N[u]\setminus \{v_{1},\ldots,v_{k'}\})$. Let $v\in N[u]\setminus \{v_{1},\ldots,v_{k'}\}$. Then $v$ can be joint to at most $\delta-k'$ vertices in $N[u]\setminus \{v_{1},\ldots,v_{k'}\}$. Thus $v$ has at least $k'$ neighbors in $S$. On the other hand, for every vertex $v$ in $S$ we have
$$|N(v)\cap S|=deg(v)-|N(v)\cap (N[u]\setminus \{v_{1},\ldots,v_{k'}\})|\geq deg(v)-\delta +k'-1\geq k.$$
Therefore $S$ is a $(k,k',1)$-dominating set in $G$. Hence,
$$
\gamma_{(k,k',1)}(G)\leq |S|=|V(G)\setminus (N[u]\setminus \{v_{1},\ldots,v_{k'}\})|=n-\delta+k'-1.
$$

\noindent {\bf Proof of 2:} Suppose that $v_{1},\ldots,v_{k+1}\in N(u)$. By assumption,
$|N[u]\setminus \{v_{1},\ldots,v_{k+1}\}|\geq 2$ and the subgraph induced by
$N[u]\setminus \{v_{1},\ldots,v_{k+1}\}$ has no isolated vertices. An argument similar to that described
in Part 1 shows that $S=V(G)\setminus (N[u]\setminus \{v_{1},\ldots,v_{k+1}\})$
is a $(k,k',1)$-dominating set in $G$. Therefore
$$
\gamma_{(k,k',1)}(G)\leq |S|=|V(G)\setminus (N[u]\setminus \{v_{1},\ldots,v_{k+1}\})|=n-\delta+k.
$$

\noindent It is easy to see that the upper bounds are sharp for the complete graph $K_{n}$, when $n\geq \max\{k,k'\}+3$.
\end{proof}
Considering Parts 1 and 2 of Theorem 3.1 we can see that
$$\gamma_{(k,k',1)}(G)\leq n-\delta+ max\{k,k'-1\},$$
when $\delta(G)\geq max\{k,k'\}+2$.\\

As an immediate consequence we conclude the following corollary.

\begin{corollary}
If $G$ is a graph of order $n$ and minimum degree $\delta\geq3$, then $\gamma_{2r}(G)\leq n-\delta+1$ and the bound is sharp.
\end{corollary}

\noindent The authors in \cite{kn} showed that $\gamma_{2r}(G)\leq n-2$ for every graph of order $n$ and minimum degree $\delta(G)\geq 3$. In fact, Corollary 3.2 gives an improvement for this bound. In addition, if $\delta\geq4$,
then the upper bound $n-2$ for $\gamma_{2r}(G)$ is not sharp.

redpajama_set_name: RedPajamaArXiv


9 out of 10 based on 759 ratings. 2,344 user reviews.
TROY-BILT 1925 OWNER'S MANUAL Pdf Download.
TROY-BILT 30237 OWNER'S MANUAL Pdf Download.
wwwalslib››Portable Generator›30237View and Download Troy-Bilt 30237 owner's manual online. BRIGGS & STRATTON. 30237 Portable Generator pdf manual download. Also for: 30237.
wwwa›Forum›Garden›TroySOURCE: My Troy-bilt generator will not generate watts. I have a Troy bilt 7800 13500 generator. It has maybe 40 hrs on it. And It stopped working. I took it into a shop to have it fixed. Repair cost would be $1000 I paid 1200 for it. Shop said the 3 main parts of the generator were fried.


Our Chicken & Cheese with sweet peppers and mushrooms.
Thinly sliced chicken breast cooked to order and served on a genuine Amoroso roll right from the City of BrotherlyLlove.
Our chicken sandwich with melted provolone and pizza sauce.
Made with fresh Boar's Head cappicola, genoa salami, mortadella, provolone cheese, lettuce, tomato and oregano served on a genuine Amoroso roll imported from Philadephia.
Sweet peppers, mushrooms and grilled onions with melted white American cheese, served on a genuine Amoroso roll imported from Philadelphia.
AKA Taylor Ham an East Coast favorite. Sliced, grilled and served on a genuine Amoroso hoagie or Kaiser roll imported from Philadephia.
Served on a genuine Amoroso Kaiser roll.
Hamburger with sweet peppers served on a genuine Amoroso Kaiser roll.
Hamburger with sauteed mushrooms served on a genuine Amoroso Kaiser roll.


A useful site offering some great rates for hotels in Bath.
Properties let weekly for your stay in Bath.
Brooks Guesthouse in Bath, is a relaxed Bed and Breakfast situated in the heart of Bath city centre. Brooks is ideally placed to offer tourists, business travellers,groups or families, excellent access to Bath and everything the city has to offer.
A charming Victorian House situated in a quiet residential area just minutes from the heart of bath.
Planning a trip to England? Prefer not to be rushed around in a large tour group? Then book a stay with us in our beautiful 18th century thatched home and enjoy your personalized tour of the Southern counties of England.
Bed and Breakfast in style at Eagle House.
A comprehensive searchable site listing B&B's throughout the UK. Simple to use.
Farm Stay is the largest network of rural and farm based accommodation in the UK. The Bath and Wiltshire Group of Farm Stay UK is an association comprising owners of holiday accommodation working together to welcome business people and tourists alike.
Harringtons Hotel Friendly, family run hotel in the heart of the City of Bath.
Hotel directory with hotels in Bath and the surrounding area. All hotels can be booked instantly and 100% securely.
Explore historic bath with the Hilton Bath City hotel. Enjoy a 2 night stay with 2 entrance tickets to the Jane Austen Centre and 2 tickets for the Bath Sightseeing bus tour from just £114 per room.
A fine Victorian house set in its own grounds.
Stylish Georgian house hotel and innovative restaurant set amongst magnificent gardens. Good connection with the Jane Austen Centre.
Hunstrete House, England's quintessential country house hotel is only a stones throw from the elegant city of Bath.
While you are in Bath why not visit the Kennard Hotel, A Georgian Town House of Charm and Character.
Marlborough House is an enchanting small hotel owned and operated by Laura and Charles Dunlap in a friendly and informal style.
Situated overlooking the Kennet and Avon Canal, close to the city centre, this modern contemporary hotel offers an ideal base when exploring this historical city.
An accommodation site for the whole of the UK and Ireland. Hotels, guest houses and self-catering.
Family owned camping and caravan site.
An accommodation site with a difference. Stay in places where historical characters have stayed.
Central hotel with reasonable prices and a lively atmosphere.
An elegant Victorian residence, built in 1876, containing a wealth of period features.
Nice site offering a range of bookable accommodation in Bath and also worldwide.
Rather luxurious self-catering holiday accommodation in Bath catering for up to 14 people.
A very special hotel. 5 stars. On the edge of the city and a lovely stroll in along a classic Georgian boulevard.
For comfort and hospitality in the heart of the countryside.
A delightful Grade 2 listed Georgian manor and farmhouse with rambling farm buildings and courtyard. Situated only 8 miles from Bath.
The Royal Crescent Hotel, the ultimate in luxury accommodation. With 45 bedrooms, the Bath House Spa and Dower House Restaurant.
Friendly hotel well situated for short breaks or a city break. This Hotel offers a hot tub, swimming pool, car park and licensed restaurant.
Bath's most exclusive townhouse hotel and sponsor of the Jane Austen Festival. Situated in Great Pulteney Street.
Enjoy newly renovated bedrooms, living areas and kitchen space all to yourself at one of our 5 unique, luxury cottages and apartments.
Widbrook Grange, home of resident owners Jane and Peter Wragg is an elegant 250 year old Georgian country house hotel peacefully located in 11 acres of grounds on the outskirts of the ancient saxon town of Bradford-on-Avon.


The Adjust Facial Features command is a light, automated version of the infamous Liquify feature, the power tool used to retouch many a magazine photo. Adjust Facial Features doesn't allow for much manual manipulation, but the effect is pretty dramatic, not to mention seamless and easy to use. Get ready for retouching requests from friends and family!
In the dialog box, a circle surrounds the area that will be affected by the adjustments. Note that Elements first analyzes whether it's truly a face or not. We tried it with an image of a cat. No dice.
Use the sliders to modify facial features such as eyes, nose, lips, chin, and face, as shown here.
The Adjust Facial Features command easily modifies facial characteristics.


Westfield Motor House is a used car dealer based in Bradford specialising in selling quality used cars to customers throughout West Yorkshire including areas such as Bradford, Leeds, Shipley, Pudsey, Halifax and Bingley.
Our aim is to help you drive away with the second hand car that suits your needs both practically and financially and we offer some of the most competitive deals on a variety of leading marques such as Audi, BMW, Citroen, Ford, Honda, Jaguar, Kia, Land Rover, Mercedes, Mini, Nissan, Peugeot, Renault, Skoda, Toyota, Vauxhall and Volkswagen to name just a few. We have a great selection of quality used cars for sale from luxury cars and large family cars to small hatchbacks and cars ideal for the first time buyer so we are confident that we can find the right car for you. View our current range in our online used car showroom.


text:
Environment chief set to meet Caraga folk in Surigao City "People's Hour"
DENR, MGB, PNA, Surigao City
Environment Secretary Horacio C. Ramos is expected to hold a "one-on-one" dialog with the different clients of the Department of Environment and Natural Resources (DENR) during the "Peoples Hour" on Wednesday, June 16, at the Surigao City Cultural Center in Surigao City.
"Secretary Ramos wants to know the issues and concerns of the people about environment in Caraga Region as he would try to find solutions on his level about their problems concerning environment" said DENR Region XIII regional executive director (RED) Edilberto S. Buiser.
"We thought there are some issues which only the Secretary of the department could solve and this is the right opportunity for the people to meet our Secretary " Buiser said.
The " People's Hour " is a regular activity of the DENR secretary, who as part of his commitment, would often go out from his office to visit the provinces and meet the people with unresolved issues concerning environment. This include unsettled land problems resulting to bureaucratic "red-tape" system of the government, mining , forestry, environmental management problems due to policies and other related concerns.
This will be the fifth time that a DENR Secretary has visited the provinces since the People's Hour was introduced by then Secretary Lito Atienza in 2008. The last time that a DENR Secretary visited a province at a People Hour was at Puerto Princesa in Palawan.
Ramos is expected to arrive at the Surigao City airport direct from Manila via the Air Philippines flight at 11:45 am on Wednesday. He will be accompanied by his technical staff, including a legal team that will work hand-in-hand with him during the People's Hour forum.
Top DENR regional officials composed of Buiser, Directors Ali Bari of Forest Management, Paquito M. Rosal of Land Management Sector, Virgilio de la Cruz of the Protected Areas and Wildlife, Coastal Zone Management Services, and the Ecosystem Research and Development Services, Ester Olavidez of the Environment Management Bureau and Alilo Ensomo of the Mines and Geo Sciences Bureau. (PNA)
DENR MGB PNA Surigao City

redpajama_set_name: RedPajamaCommonCrawl


text:
{{BasketCount}}
{{=BasketCount}}
{{/BasketCount}} {{IsAuthenticated}}
{{:IsAuthenticated}}
{{/IsAuthenticated}}
Bus Tracker
Tickets & Fares
Coach and Bus Hire
Snap of the Month 2022 05/01/2022
Are you a keen photographer, a bus enthusiast or just fancy taking part in our Snap of The Month 2022? We have a fun, outdoor activity for you, competing to be the NEXT BEST PHOTO!!
It's an opportunity to feature your wonderful photo, if chosen, to appear on our website and our social media in front of thousands of viewers.
How to enter...
Each month we will issue a new keyword - the monthly keywords are your photo inspiration... we want to see some creative ways of taking photos of our buses, our brand or something in relation to us as a business (Bus stop flags, posters, drivers - if they're happy for the photo!)
Buses at stops or on the move whilst in the rain or sunny skies, a city shot with one of the buses in the background, a glimpse as a back seat passenger, ariel shots or even a selfie with our logo - whatever you think will make a wonderful photo!
*** Find this month's keyword on our social channels (Click the socials below)***
Twitter / Facebook / Instagram
Send in your photos each month to us...
Tag us on socials with the hashtag #SnapOfTheMonth or email them directly to Marketing@AdventureTravel.Cymru
The winner's photo will be displayed on the 28th of each month, for one month. December 2022 will feature all winner photos together again!
We reserve the right to use your photo/s in any of our marketing material for the future.
Please be mindful of other passengers and people as you take photos, please ask permission should you take anyone's photo. Thank you.
Modern Slavery and Human Trafficking Policy 2019 13/05/2021
We are committed to improving our practices to combat slavery and human trafficking.....
NAT Group Announces New Night Buses in Cardiff 26/07/2019
LEADING Welsh bus operator NAT Group has announced an extension to its X1 and X8 bus routes
Press Release August 2019 30/07/2019
All change please! NAT Group named new operator for key Swansea bus routes, and simplified fares!
All Aboard! NAT Group fundraising for Children In Need 30/07/2019
NAT Group fundraises for BBC Children in Need.
New Cardiff to London coach service launched 21/11/2019
A south Wales bus company has joined forces with a national intercity technology platform to launch a new Cardiff to London coach service, a…
Adventure Friday 28/11/2019
It's Adventure Friday, like Black Friday but Better!
Swansea Park & Ride Starts 1st December 29/11/2019
We are pleased to announce that the Sunday 2019 Christmas Park & Ride service will be operated by NAT Group.
Bus Pass Renewal Programme 05/12/2019
Transport for Wales is currently managing the reissue of free bus passes across Wales. Here's an update on the renewal programme we're manag…
Gower Service Changes 23/12/2019
Changes to the 115 and 119 services in Swansea from Monday 6th January 2020.
815 Cancellation 16/01/2020
815 Service Cancellation from Friday 21st Feb 2020
NAT Group Goes the Extra Mile for Flood-Hit Pontypridd 21/02/2020
Following the devastating floods across South Wales this week, local bus and coach operator NAT Group has pledged to support homes and busin…
FIVE and X5 Service Update 11/03/2020
With regret, we announce that routes FIVE and X5 will be ending in May 2020, with the last day of operation being Saturday May 2nd.
TrawsCymru Weekend Free Travel 31/03/2020
Important changes to the TrawsCymru Weekend Free Travel in light of the COVID-19 outbreak.
NAT Group goes above and beyond to ensure maximum compliance 17/04/2020
LEADING Welsh bus operator NAT Group has partnered with a local tachograph specialist to monitor all compliance across its drivers and 150-v…
Passengers show support to public transport key workers 17/04/2020
Prominent Welsh bus operator, NAT Group, has received messages of gratitude from customers for its services during the Covid-19 pandemic.
International Travel Annoucement 18/05/2020
With international travel all but banned for non-essential trips, NAT Group has announced it will discontinue its overseas operations and fo…
Essential link service between Gorseinon and Pontarddulais 28/05/2020
We are pleased to work with Swansea City Council to ensure the essential link between Gorseinon and Pontarddulais…
NAT Group to continue the 'Severn Express' service between Bristol & Chepstow 10/06/2020
We are delighted to announce that NAT Group, in partnership with TrawsCymru and Monmouthshire Council, have unveiled plans to continue the '…
Cardiff City Center Service Diversions From 21st of June 19/06/2020
Due to road closures in Cardiff city center from midday Sunday the 21st of June, following the recent announcement by Mark Drakeford...
Cardiff School Transport Annoucement 22/06/2020
We are pleased to confirm that our 800-series (801, 802 etc) Cardiff school services WILL operate from June 29th until the end of the academ…
fflecsi - Cardiff's first 'Demand Response Transport' scheme 25/06/2020
Cardiff is set to trial its very first Demand Response Transport (DRT) scheme where passengers can remotely book bus journeys, and schedules…
Face Coverings 23/07/2020
Following a consultation with Welsh Govt, we can now confirm that face coverings will be mandatory on buses from the 27th July 2020. Our dri…
School Bus Passes 2020/21 23/07/2020
Following lengthy and detailed conversations with Cardiff Council and reviewing changes with regards to public transport in Wales, we are no…
New service launch in Cardiff, the 'C2' 29/07/2020
In September we will be launching a brand-new bus route connecting Capital Retail Park and Cardiff City Stadium with Pengam Green (Tesco) vi…
X8 Service to be extended & renamed from the 7th of September 30/07/2020
From the 7th of September, we will be renaming and extending our current X8 route as part of our strategic network plans.
Bank Holiday Service Levels 31st of August 2020 20/08/2020
For our Bank Holiday service levels on Monday the 31st of August, please click here.
Adventure somewhere different for the day! 07/09/2020
For many people, 2020 has not been a good year so far and we'd like to do our bit to help people have a very reasonably priced day out and a…
NAT Group Teams up with South Wales Chamber of Commerce to Become a Corporate Partner 10/09/2020
Leading Welsh transport provider NAT Group has joined South Wales Chamber of Commerce as one of their select number of Corporate Partners, t…
Changes to Pontypridd Services 30/09/2020
We are pleased to announce that from Monday 5th October...
Service 16 - Railway Bridge Closure 06/11/2020
Due to the closure of the railway bridge at Waungron, the evening service 16 operating between Pontarddulais, Grovesend, and Gorseinon, and …
C8 Route Change 30/11/2020
Effective Sunday 6th December the C8 will operate on its new route around the City...
Christmas Jumper Day for Save the Children 10/12/2020
This year we will be running a Christmas jumper day on Friday 11th December to raise some much-needed funds for 'Save the Children'.
X7 TrawsHafren Service Update 17/12/2020
As we reported back in mid-2020, our business was successful in securing a pilot scheme for route X7...
901 Christmas Timetable 18/12/2020
901 Railinc Christmas Service Levels...
900 Bus Service 22/12/2020
Check out our Blackwood Interchange to Blackwood Gate Retail Park service...
Day in the life of a coach driver 08/01/2021
It is always an early start, we quietly shuffle out of the house, trying not to wake everybody up before the larks are even awake...
Charity of The Year 2021 21/01/2021
Adventure Travel is delighted to announce their Charity of The Year 2021...
304 Diversion - Emergency road closure in Barry 11.02.21 10/02/2021
The emergency road closure will be taking place on 11.02.21 from 1830 to 2300 hours.
Our 304 INBOUND to Cardiff will be affected, a slight …
Day in the life of a bus driver: February 17/02/2021
What a curious month February has been...
303 and 304 roadwork diversions 03/03/2021
Please be aware over the next coming weeks, our 303 and 304 bus services in St Athan will be disrupted due to road closures...
Covid-19 FAQ 08/01/2021
DONATE & WIN this 2021 signed welsh rugby jersey from WRU! 08/03/2021
We are delighted to donate this 2021 signed welsh rugby jersey from WRU to raise money for our Charity of The Year 2021 - Tŷ Hafan ...
International Women's Day 2021 - Cerianne's story 05/03/2021
Fast forward to 2021, we are looking much closer to home - the transport industry. Well known for its male-dominated world, we acknowledge a…
Easter Service Levels 22/03/2021
The following service levels will in operation over the Easter period...
Direct routes to Mass Vaccination Centre Cardiff 24/03/2021
We're pleased to confirm that Adventure Travel will be running additional journeys from the 28th of March to the Mass Vaccination Centre in …
April blog - Training School at Adventure Travel 30/03/2021
"We could soon be in the top 5 training schools in Wales..."
Website Maintenance update - May 5th 2021 05/05/2021
We apologise for any inconvenience caused this evening...
Scheduled Maintenance 05/05/2021
Sorry for the inconvenience but we're performing some scheduled maintenance this evening.
To find out more see here.
2021 Bank Holiday Service Levels 23/04/2021
The following service levels will in operation over the May bank holiday period...
May blog - Cleaning at Adventure Travel 10/05/2021
Cleaning is and always has been a crucial element of our business...
Powys Services - Aberdulais Gyratory Closure 20/05/2021
Important service information regarding the T6, 62, 64, and T6X services.
Happy Fathers Day! - June blog special 10/05/2021
Happy Fathers day today!
We have a special blog this month, where we talk to two of our drivers - a father and son duo...
Driver writes a children's book about buses! 05/07/2021
A driver has written and self-published a children's book called 'Burlington Buses'...
Corona Virus Service Updates - Updated 04.05.2021 11/03/2021
Please click here for the latest changes and updates to our services during Covid-19.
Adventure to Porthcawl this summer with our 404! 02/06/2021
We're extending our 404 bus service to provide a direct route to the popular seaside town of Porthcawl...
Cardiff Match Day 10th July 2021 29/06/2021
Closure and Diversion information for Saturday 10th July 2021
July blog with Adam Keen, Managing Director 23/06/2021
In this month's blog, we're featuring our managing Director Adam which includes his thoughts on the pandemic and the future of Adventure Tra…
A day in the life of... a bus inspector! 04/08/2021
Let us introduce you to Dai, who is our Revenue and Standards Inspector at Adventure travel...
Adventure Travel – A 2021 update from your bus operator 28/07/2021
Start your engines!
After several 'false starts' during 2020, a return to routine and normality is finally upon us and we can restart the …
2021/22 School Bus Passes 16/08/2021
Our 2021/22 School Bus Passes are released. See the article for further information.
Elvis is coming to Wales this month! 02/09/2021
We're delighted and very excited to announce the Elvis festival will be going ahead in Porthcawl this month!
404 - It's Porthcawl Extension is staying 20/10/2021
We are pleased to announce that the 404 extension to Porthcawl will now be an all-year extension...
A day in the life of... a school bus driver! 14/09/2021
Our September blog is dedicated to our school bus drivers, back behind the wheel, ensuring safe journeys to our local schools...
The mechanics of the company... the Engineering team! 01/11/2021
Chris, our Head of Engineering has given us a great insight into his position within Adventure Travel and how the Engineering department wor…
An eight-week consultation on Cardiff's new Bus Strategy 25/10/2021
A new strategy, which aims to double the number of people who use buses in Cardiff, reduce congestion and improve air quality, is out now fo…
Road safety week 2021 (15th November to 21st November) 11/11/2021
Welcome to ROAD SAFETY WEEK, inside November's blog we discuss what actions we take at Adventure Travel to ensure all our drivers and passen…
Cardiff Council £1 travel scheme 02/12/2021
Cardiff launches ten days of £1 bus fares in the run-up to Christmas...
for further details and the boundary map, please click here.
Pontypridd Christmas Service Levels 2021 30/11/2021
Swansea Christmas Service Levels 2021 30/11/2021
Cardiff Christmas Service Levels 2021 30/11/2021
Feedback Drives Upgrades for Flagship X1 Bus Service 18/10/2021
Adventure Travel is enhancing its flagship X1 Culverhouse Cross - Pontprennau bus route
Free Bus Travel in Swansea - February 2022 30/07/2021
We've teamed up again with Swansea Council to provide Free bus travel in Swansea...
Merry Christmas from Adventure Travel! 20/12/2021
From our team across South Wales, we wish you and your family a very merry Christmas!
Coach Service
Advertise on our buses
Tutorial - Bus Tracker
Tutorial - Journey planner
Tutorial - Ticket Gifting
© 2019 NAT Group Ltd - Powered by BusHub Mobility Platform.