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In [[graph theory]], the '''graph bandwidth problem''' is to label the ''n'' vertices ''v<sub>i</sub>'' of a graph ''G'' with distinct integers ''f''(''v<sub>i</sub>'') so that the quantity <math>\max\{\,| f(v_i) - f(v_j)| : v_iv_j \in E \,\}</math> is minimized (''E'' is the edge set of ''G'').<ref>{{harv|Chinn|Chvátalová|Dewdney|Gibbs|1982}}</ref>
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It has a [http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=%0D%0A%0D%0AThere+are+hundreds+of+multi-tools+out+there+and+dozens+of+quality+ones%2C+so+how+do+you+go+about+deciding+upon+the+proper+a+single+for+your+outside+adventure%3F+I+hope+that+I+provided+some+beneficial+insights+right+here%2C+but+I+can+not+tell+which+tool+is+the+finest+for+you.+I%27ve+constantly+place+tools+to+the+test+by+pushing+them+to+the+point+of+breaking%2C+but+practically+nothing+has+ever+stood+up+to+my+every+day+use%E2%80%A6Except+the+Kolbalt+Hammer%2C+a+low+cost+Stanley+Flat+Pry+bar+and+an+actual+crowbar%E2%80%A6Hell+not+even+Craftman+have+withstood+my+every+day+use%E2%80%A6.+This+might+be+a+good+tool+for+some+men+and+women+if+not+a+majority%2C+but+as+for+me%2C+I%27m+not+a+fan%E2%80%A6At+least+not+until+I+come+across+a+single+that+can+manage+the+tension+of+everyday+use%E2%80%A6.+It%27s+been+a+long+time+because+a+multi+tool+has+moved+on+from+becoming+perceived+as+a+clunky+piece+of+steel+on+a+farmer%27s+belt+utilised+only+to+reduce+fence.+To+be+fast+sufficient%2C+you+would+have+to+make+the+initial+cuts+employing+the+tool+and+then+tear+the+pants.%0D%0A%0D%0AIt+may+also+be+easy+for+other+men+and+women+to+think+about+purchasing+tools+which+have+removable+components+just+in+case+some+of+them+are+not+working+effectively.+General%2C+obtaining+a+multitool+around+is+a+terrific+and+intelligent+way+for+men+and+females+alike+to+prepare+for+any+sort+of+scenario.++Here%27s+more+information+in+regards+to+%5Bhttp%3A%2F%2Fwww.eletromaniacos.com%2FWebid%2Fitem.php%3Fid%3D88998+ridgid+jobmax+multi+tool+review%5D+review+our+web-site.+Though+it+all+boils+down+to+preference%2C+realizing+the+distinction+between+a+high+high-quality+tool+and+a+poorly+developed+one+particular+could+figure+out+one%27s+survival+in+preparing+for+the+worst+doable+circumstance.+With+that+said%2C+you+need+to+also+use+the+triangular+grout+removal+tool+that+can+be+found+at+Lowe%27s.+Trust+me%2C+I+know+firsthand+that+the+Bosch+multi+tool+is+expensive%2C+but+if+you+are+really+serious+about+removing+grout+and+undertaking+other+handyman+stuff+you+will+adore+it.+You+can+obtain+manual+grout+removal+tools+for+as+small+at+%2410.+Add+a+protective+layer+of+duct+tape+to+the+wall+tile+in+case+you+do+bump+it+with+the+Bosch+multi+tool.+Please+appear+at+for+far+more+tools.%0D%0A%0D%0AFor+the+reason+that+none+of+the+brands+packages+their+tool+the+very+same+way+it+really+is+fairly+really+hard+to+%5Bhttp%3A%2F%2Fasgaard.pf-control.de%2Findex.php%3Fsite%3Dprofile%26id%3D192+evaluate+fees%5D.+When+you+think+about+the+price+point+of+this+tool+lots+of+will+likely+discover+this+to+be+their+top+rated+option.+I+am+fairly+confident+that+this+tool+will+execute+nicely+on+the+job-site+and+it+really+is+certainly+backed+by+a+powerful+history+of+efficiency.+We%27d+appreciate+to+see+FEIN+step+this+tool+up+with+a+bigger+battery+platform+to+get+it+on+par+with+the+other+18V+models.+Nevertheless%2C+this+tool+is+an+exceptional+choice+for+DIY%27ers%2C+occasional+PRO+use%2C+and+users+that+already+have+the+Porter-Cable+20V+battery+platform+when+you+take+into+consideration+the+superb+value+point+of+the+bare+tool.+We+never+ever+have+and+never+ever+will+accept+payment+in+exchange+for+a+good+overview.+Each+tool+is+labeled+on+the+handles.%0D%0A%0D%0AI+use+it+for+fishing+%28as+its+advertised%29+but+I+also+hold+it+with+me+as+a+versatile+pocket+knife+for+the+purpose+that+of+its+smaller+sized+sized+size%2C+fundamental+sleak+disign+and+it+is+non+intrusive+aesthetics.+I+don%27t+like+multi-tools+%28attempt+installing+a+water+bottle+cage+with+one%29+so+I+carry+the+hex+L-wrenches+and+open+end+wrenches+necessary%2C+along+with+a+chain+tool+and+some+swift+links+wrapped+in+a+shop+rag%2C+alternatively.+I+often+carry+a+Park+Tool+IB-3+%5Bhttp%3A%2F%2Fseyffenstein.Mikronation.de%2Fwiki%2Findex.php%3Ftitle%3DBest_Military_Multi_Tool+multi+tool%5D+which+offers+1.5mm%2C+2mm%2C+two.5mm%2C+3mm%2C+4mm%2C+5mm%2C+6mm%2C+and+8mm+hex+wrenches%2C+a+flat+blade+screwdriver%2C+a+T25+Star+driver+for+disc+brake+rotor+screws%2C+an+fantastic+composite+wrapped+tire+lever+which+triples+as+a+spoke+wrench+of+two+sizes+%283.23mm+%26+3.45mm+nipples%29%2C+as+properly+as+serves+as+the+deal+with+for+tightening+its+7+thru+ten-speed+capable+chain+breaking+tool.++This+multi+tool+weighs+just+6+ounces+%28170+grams%29+per+the+link+supplied+below.+Carrying+around+a+load+of+tools+on+the+bike+is+a+discomfort.%0D%0A%0D%0ARegrettably%2C+when+I+try+to+tell+you+the+name+of+the+knife%2C+your+mediocre+site+form+says+the+word+is+%22obscene%22+and+won%27t+let+me+post+it.+You+can+get+in+touch+with+it+a+%22kukri+machete%22+if+you+want+to%2C+considering+the+fact+that+it+is+typically+marketed+that+way%2C+but+it%27s+just+a+kukri%2C+which+is+a+multi-goal+knife+%28as+described%29+created+for+and+made+use+of+by+Gurkha+warriors+and+Nepalese+farmers.+It+is+a+Nepalese+knife+design+and+style%2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titanium body] lock design that combines simplicity with sturdiness and a blade created from S35VN stainless-steel that without end retains its edge. As soon as opened and locked the knife feels as solid as a set blade and holds comfortably in the hand. The quality is really second to none and it’ll shave the hairs on your chin proper out of the field!<br><br>The large hole in the deal with serves the same purpose as a thumb stud, to allow for one handed opening. The body is a highly textured nylon that is quite stiff. The rear of the knife has a lanyard hole. On the latest models they have improved the pocket clip to make it ambidextrous, so you [http://www.knife-depot.com/pocket-knives/ kershaw pocket knife] can mount it to both facet of the knife. They have also upgraded the blade to a slightly fancier stainless steel. There have been multiple small modifications over the years, as they preserve making small tweaks to the identical fundamental design.<br><br>In addition to its single handed operation, what made the Christy knife so useful was its blade high quality. You'll be able to tell that Christy has a long historical past of creating scalpels and razor blades, because for me this has at all times been an exceedingly sharp knife. Though the Christy knife is designed for easy blade substitute, I've [http://www.thebestpocketknifereviews.com/best-multi-tool-survival-pocket-tool-review/ Best Camping Multi tool] had the identical blade in mine for the last ten years, and I contact up the sting solely sometimes with a ceramic hone. I like that the Christy knife stays 100% American made and backed by a typical sense lifetime guarantee which does not cowl re-sharpening or abuse, but does cover some other defects.<br><br>It isn't good to use ceramic, marble, or glass boards because of their hard floor. When the perimeters of the knife hit the laborious board, it could possibly get boring easily. When you love doing things within the kitchen or in case you are a professional chef, you should take your time in purchasing for the most effective professional knife sharpener. Due to the web, you may simply make the appropriate pick. You need to think about your wants and the cash that you just're willing to spend. Chef's Choice is a superb choice, together with other outstanding knife sharpener brands. Look around and discover the best deal.<br><br>This category describes use round the home, the occasional tenting trip, or an occasional carry knife. That is the most versatile class, with the widest range of knives to satisfy this type of use. You don’t have to fret so much about things like weight or size, as a result of you'll only carry it as you need it. The final and essential consideration for buying a tactical folding knife is to factor in the toughness of the supplies and the manufacturing quality. These knives usually receive a number of abuse, and want to be able to maintain up to vigorous on a regular basis put on and tear without failure the components.<br><br>At present the company nonetheless lives on even stronger than it was five or ten years in the past in the same town the place it was founded with a new location and an updated website. The Christy Sliding Blade Pocket Knives are still made on the unique 53-operation production line and gear, which is now operated and produced by machinist Hal R. Christy, son of Earl Christy. There are very few fourth-generation, family owned corporations on this planet, and this one continues to offer “the most convenient pocket knife ever designed” out there in the Companion (nickel plated), [http://Procamp.Uni-Leipzig.de/doku.php?id=profile_grdalrympl Executive] (polished black nickel plated), and the Traveler (matte end Teflon nickel plated). Features & perform<br><br>SOG is a pacesetter in knife manufacturing and their knives have won praise with many in the business. The SOG Flash II is a superb affordable choice for the each-day-carry knife and appears similar in appearance to the Benchmade Griptilian. It’s razor sharp with an AUS-eight stainless-steel blade which is partially serrated and really durable. The Flash II uses [http://www.thebestpocketknifereviews.com/best-multi-tool-survival-pocket-tool-review/ Www.thebestpocketknifereviews.Com] spring assisted opening know-how which ensures a quick, easy opening and the handle is glass-reinforced nylon which leads to a light-weight but strong design. For added safety the knife includes a [http://sss.chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=Oscillating+Multi-Tools+%28OMT%29+have+%5Bhttp%3A%2F%2FEsmikorea.ezconn.net%2Fxe%2F%3Fdocument_srl%3D70024+exploded%5D+in+recognition+and+produced+a+substantial+sector+of+the+power+tool+market.+These+%5Bhttp%3A%2F%2FAdvisory-Council.mol.go.th%2Fnode%2F33327+versatile+tools%5D+give+contractors%2C+renovators+and+DIY%27ers+a+go-to+alternative+for+a+variety+of+cutting%2C+grinding+and+sanding+applications+where+restricted+space+is+an+concern.++If+you+want+to+learn+more+regarding+%5Bhttp%3A%2F%2Fwww.thebestpocketknifereviews.com%2Fbest-multi-tool-survival-pocket-tool-review%2F+Best+Survival+Multi+Tool%5D+check+out+our+webpage.+Current+developments+in+battery+technology+have+allowed+the+tool+producers+to+create+cordless+OMT%27s+with+equivalent+energy+and+overall+performance+to+the+original+corded+versions.%0D%0A%0D%0A%0D%0A%0D%0AThe+Leatherman+Skeletool+is+my+choice+for+a+little+multitool.+Yes%2C+it%27s+four%22-closed%2C+but+it%27s+truly+slim+and+light.+With+its+pocket+clip%2C+it+carries+like+a+regular+pocket+knife.+Every+portion+is+solid+stainless+steel+or+reinforced+with+stainless.+This+higher+grade+metal+will+not+rust+even+when+immersed+in+water.+Leatherman+provides+a+25+year+warranty+on+all+its+items.+The+tool+is+4+inches+long+when+closed+and+weighs+approximately+8.5+ounces.+%28This+a+single+is+my+personal+favorite%21%29+The+pliers+are+a+compound+tool+and+exert+greater+mechanical+leverage+than+standard+hand+stress.+The+knife+is+modest+but+serrated+and+hugely+versatile.+The+saw+is+fine+and+difficult+sufficient+to+cut+by+means+of+copper+pipe.+Stainless+Steel+Locking+Multitool+Pliers+w%2F+Sheath+831102+Grout+removal+blades%0D%0A%0D%0AWe+also+wanted+to+mention+Victorinox%27s+SwissTool+%28from+the+makers+of+the+popular+Swiss+Army+Knife%29+and+CyberTool+%2C+which+many+of+you+nominated+for+the+reason+that+of+Victorinox%27s+reputation+and+history+for+nicely+crafted%2C+effectively+constructed+solutions.+Each+the+CyberTool+and+SwissTool+are+product+lines%2C+so+you+can+pick+the+multitool+that+has+the+quantity+of+person+tools+and+the+type+issue+that+you+happen+to+be+looking+for+and+operates+in+your+price+range.+The+SwissTool+is+commonly+regarded+as+a+far+more+all-goal%2C+multi-purpose+tool%2C+when+the+CyberTool+has+features+that+are+more+geared+towards+somebody+who%27ll+have+to+have+to+do+repairs+about+the+office+or+with+technology%E2%80%94like+the+inside+of+your+household-constructed+Computer.%0D%0A%0D%0AIt+is+notable+that+this+is+the+only+tool+in+this+group+that+does+not+use+an+18V+battery%2C+this+tool+runs+on+a+14.4V+platform.+This+is+yet+another+1+of+the+tools+that+does+not+function+a+work+light+in+it+is+style.+The+FEIN+AFMM+14+does+give+a+tool-absolutely+free+blade+change+%28FEIN+is+the+father+of+these+tools+and+the+tool-totally+free+design%21%29%2C+along+with+a+typical+on%2Foff+switch+and+variable+speed+dial.+The+tool-no+cost+blade+adjust+technique+uses+a+removable+pin%2Fbolt+that+holds+the+blades+in+position.%0D%0A%0D%0AThe+tools+are+accessible+devoid+of+obtaining+to+unfold+the+whole+tool+which+to+me+is+handy+when+attempting+to+do+factors+1+handed.+I+often+handle+to+start+off+factors+and+get+a+single+hand+busy+and+then+hunt+for+my+tools.+I+have+to+use+tools+that+match+my+impatient+nature+sometime+because+I+start+off+issues+and+then+have+to+catch+up+by+grabbing+a+tool+with+1+hand.+You+can+also+study+far+more+information+about+Gerber+22-41545+Black+Diesel+Multi-Plier+with+Sheath+tool+as-nicely.+I+did+start+out+this+project+applying+my+Dremel+Multi-Max+but+it+died+following+10+minutes.+This+truly+upset+me+considering+that+I+just+opened+and+applied+a+new+grout+attachment+%28which+expense+%2425%29.+Arrrrrrrgggghhhhhh.+Emergency+Travel+Tool+22-01769+Stainless+Steel+Tools++Pocketknife+Critique+SOG+Flash+Rescue+%2467.50+What+multitool+do+you+carry%3F%0D%0A%0D%0AYeah%2C+the+squirt+is+significantly+smaller+sized+than+it+appears+on+that+page.+I+also+second+the+Juice.+1+issue+with+the+Juice%2C+at+least+my+version%2C+is+that+the+flathead+blades+are+incredibly+complicated+to+pull+out.+They+are+jammed+in+there+fantastic.+A+multi-tool+might+not+replace+a+fantastic+toolbox%2C+but+I%27d+rather+carry+it+around.+A+knife+crafted+from+420HC%2C+a+higher-carbon+stainless+steel+that+is+corrosion+resistant+and+can+be+quickly+maintained.+A+stainless+steel+blank+coated+with+premium+synthetic+diamond+particles+which+develop+a+rough+surface+that+can+be+made+use+of+to+sharpen+difficult+steel+blades+with+a+great+deal+far+more+precision+than+wood%2Fmetal+files.+In+terms+of+price+tag%2C+we%27ve+seen+the+Gerber+600+retailed+for+roughly+%2440%2C+when+the+Leatherman+Wave+retails+for+around+%2460.%0D%0A%0D%0AI%27m+a+tool+guy.+I+always+have+been.+When+I+was+a+kid+I+was+notorious+for+taking+items+apart+and+%28often%29+having+them+back+with+each+other+once+again.+I+was+normally+developing+items+with+saws%2C+hammers%2C+drills%2C+nails+making+use+of+the+occasional+cost-free+sheet+of+plywood+that+would+fall+off+a+truck.+My+enjoy+of+tools+also+applies+to%2C+of+course%2C+hiking+and+backpacking.+What+follows+is+an+report+I%27m+positive+all+backpackers+can+identify+with+in+a+single+way+or+one+more.+Hopefully+my+recommendations+on+the+appropriate+option+for+backpacking+tools+make+sense+to+you+as+considerably+as+it+does+me.+My+Burmese+Machete&create=Create locking swap] which will ensure the blade does not accidentally deploy.
The problem may be visualized as placing the vertices of a graph at distinct integer points along the ''x''-axis so that the length of the longest edge is minimized. Such placement is called '''linear graph arrangement''', '''linear graph layout''' or '''linear graph placement'''.<ref name=feige/>
 
The '''weighted graph bandwidth problem''' is a generalization wherein the edges are assigned weights ''w<sub>ij</sub>'' and the [[Loss function|cost function]] to be minimized is <math>\max\{\, w_{ij} |f(v_i) - f(v_j)| : v_iv_j \in E \,\}</math>.
 
In terms of matrices, the (unweighted) graph bandwidth is the [[bandwidth (matrix theory)|bandwidth]] of the [[symmetric matrix]] which is the [[adjacency matrix]] of the graph.
 
==Bandwidth formulas for some graphs==
For several families of graphs, the bandwidth <math>\varphi(G)</math> is given by an explicit formula.
 
The bandwidth of a [[path graph]] <math>P_n</math> on n vertices is <math>1</math>, and for a complete graph <math>K_m</math> we have <math>\varphi(K_n)=n-1</math>. For the [[complete bipartite graph]] <math>K_{m,n},</math>
:<math>\varphi(K_{m,n}) = \lfloor (m-1)/2\rfloor+n</math>, assuming <math>m \ge n\ge 1</math>,
which was proved by Chvátal.<ref>A remark on a problem of Harary. V. Chvátal, ''Czechoslovak Mathematical Journal'' '''20'''(1):109&ndash;111, 1970. [http://dml.cz/dmlcz/100949 http://dml.cz/dmlcz/100949]</ref> As a special case of this formula, the [[star graph]] <math>S_k = K_{k,1}</math> on ''k''+1 vertices has bandwidth <math>\varphi(S_{k}) = \lfloor (k-1)/2\rfloor+1</math>.
 
For the [[hypercube graph]] <math>Q_n</math> on <math>2^n</math> vertices the bandwidth was determined by {{harvtxt|Harper|1966}} to be
:<math>\varphi(Q_n)=\sum_{m=0}^{n-1} \binom{m}{\lfloor m/2\rfloor}.</math>
 
Chvatálová showed<ref>Optimal Labelling of a product of two paths. J. Chvatálová, ''Discrete Mathematics'' '''11''', 249&ndash;253, 1975.</ref> that the bandwidth of the <math>(m \times n)</math> [[lattice graph|square grid graph]] <math>P_m \times P_n</math>, that is, the [[Cartesian product of graphs|Cartesian product]] of two path graphs on <math>m</math> and <math>n</math> vertices, is equal to <math>\min\{m,n\}</math>.
 
==Bounds==
 
The bandwidth of a graph can be bounded in terms of various other graph parameters. For instance, letting χ(''G'') denote the [[chromatic number]] of ''G'',
:φ(''G'') ≥ χ(''G'')-1;
letting diam(''G'') denote the [[diameter (graph theory)|diameter]] of ''G'', the following inequalities hold:<ref>{{harvnb|Chinn|Chvátalová|Dewdney|Gibbs|1982}}</ref>
:<math>\lceil (n-1)/\mathrm{diam}(G) \rceil \le \varphi(G) \le n - \mathrm{diam}(G)</math>,
where <math>n</math> is the number of vertices in <math>G</math>.
 
If a graph <math>G</math> has bandwidth <math>k</math>, then its [[pathwidth]] is at most <math>k</math> {{harv|Kaplan|Shamir|1996}}, and its [[tree-depth]] is at most <math>k\log(n/k)</math> {{harv|Gruber|2012}}. In contrast, as noted in the previous section, the star graph <math>S_k</math>, a structurally
very simple example of a [[tree (graph theory)|tree]], has comparatively large bandwidth. Observe that the [[pathwidth]] of <math>S_k</math> is 1, and its tree-depth is 2.
 
Some graph families of bounded degree have sublinear bandwidth: {{harvtxt|Chung|1988}} proved that if ''T'' is a tree of maximum degree at most ''∆'', then
:<math>\varphi(T) \le 5n / \log_\Delta n</math>.
More generally, for [[planar graph]]s of bounded maximum degree at most ''∆'', a similar bound holds (cf. {{harvnb|Böttcher|Pruessmann|Taraz|Würfl|2010}}):
:<math>\varphi(G) \le 20n / \log_\Delta n </math>.
 
==Computing the bandwidth==
Both the unweighted and weighted versions are special cases of the [[quadratic bottleneck assignment problem]].
The bandwidth problem is [[NP-hard]], even for some special cases.<ref>Garey–Johnson: GT40</ref> Regarding the existence of efficient
[[approximation algorithm]]s, it is known that the bandwidth is [[hardness of approximation|NP-hard to approximate]] within any constant, and this even holds when the input graphs are restricted to [[caterpillar tree]]s {{harv|Dubey|Feige|Unger|2010}}. On the other hand, a number of polynomially-solvable special cases are known.<ref name=feige>"Coping with the NP-Hardness of the Graph Bandwidth Problem", Uriel Feige, ''[[Lecture Notes in Computer Science]]'', Volume 1851, 2000, pp. 129-145, {{doi|10.1007/3-540-44985-X_2}}</ref> A [[heuristic]] algorithm for obtaining linear graph layouts of low bandwidth is the [[Cuthill–McKee algorithm]].
 
==Applications==
The interest in this problem comes from some application areas.
 
One area is [[sparse matrix]]/[[band matrix]] handling, and general algorithms from this area, such as [[Cuthill–McKee algorithm]], may be applied to find approximate solutions for the graph bandwidth problem.
 
Another application domain is in [[electronic design automation]]. In [[standard cell]] design methodology, typically standard cells have the same height, and their [[placement (EDA)|placement]] is arranged in a number of rows. In this context, graph bandwidth problem models the problem of placement of a set of standard cells in a singe row with the goal of minimizing the maximal [[propagation delay]] (which is assumed to be proportional to wire length).
 
==See also==
*[[Pathwidth]], a different NP-complete optimization problem involving linear layouts of graphs.
 
==References==
{{reflist}}
*{{cite doi|10.1016/j.ejc.2009.10.010}}
*{{cite doi|10.1002/jgt.3190060302}}
*{{cite
| last = Chung
| first = Fan R. K.
| author-link =
| last2 =
| first2 =
| author2-link =
| year = 1988
| date =
| publication-date =
| contribution = Labelings of Graphs
| contribution-url =
| editor-last =
| editor-first =
| editor-link =
| editor2-last = Beineke
| editor2-first = Lowell W.
| editor2-link =
| title = Selected Topics in Graph Theory
| edition =
| series =
| place =
| publication-place =
| publisher = Academic Press
| volume =
| pages = 151–168
| id =
| isbn = 978-0-12-086203-0
| doi =
| oclc =
| url = http://www.math.ucsd.edu/~fan/mypaps/fanpap/86log.PDF
}}
*{{cite doi|10.1016/j.jcss.2010.06.006}}
* {{cite book
| last = Garey
| first = M.R.
| authorlink = Michael Garey
| coauthors = [[David S. Johnson|Johnson, D.S.]]
| title = [[Computers and Intractability: A Guide to the Theory of NP-Completeness]]
| year = 1979
| publisher = W.H. Freeman
| location = New York
| isbn =  0-7167-1045-5
}}
*{{Citation
| title = On Balanced Separators, Treewidth, and Cycle Rank
| year = 2012
| author = Gruber, Hermann
| journal = Journal of Combinatorics
| pages = 669–682
| volume = 3
| issue = 4
}}
*{{cite doi|10.1016/S0021-9800(66)80059-5}}
*{{Citation
| title = Pathwidth, bandwidth, and completion problems to proper interval graphs with small cliques
| year = 1996
| journal = SIAM Journal on Computing
| pages = 540–561
| volume = 25
| last1 = Kaplan    | first1 =  Haim
| last2 =  Shamir    | first2 =  Ron    }}
*
 
==External links==
*[http://www.csc.kth.se/~viggo/wwwcompendium/node53.html Minimum bandwidth problem], in: Pierluigi Crescenzi and Viggo Kann (eds.), ''A compendium of NP optimization problems.'' Accessed May 26, 2010.
 
[[Category:Graph algorithms]]
[[Category:Combinatorial optimization]]
[[Category:NP-hard problems]]
[[Category:Graph invariants]]

Latest revision as of 16:48, 7 January 2015

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