Abstract
“Domination in graphs” is an area of graph theory that has received a lot of attention in recent years. It is reasonable to believe that “domination in graphs” has its origin in “chessboard domination.” The “queen domination” problem asks: What is the minimum number of queens required to be placed on an 8 ×8 chessboard so that every square not occupied by any of these queens will be dominated (that is, can be attacked) by one of these queens? Recall that a queen can move horizontally, vertically, and diagonally on the chessboard. The answer to the above question is 5. Figure 10.1 gives one set of dominating locations for the five queens.
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References
Adiga, C., Balakrishnan, R., So, W.: The skew energy of a digraph. Linear Algebra Appl. 432, 1825–1835 (2010)
Aharoni, R., Szabó, T.: Vizing’s conjecture for chordal graphs. Discrete Math. 309(6), 1766–1768 (2009)
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, MA (1974)
Akbari, S., Moazami, F., Zare, S.: Kneser graphs and their complements are hyperenergetic. MATCH Commun. Math. Comput. Chem. 61, 361–368 (2009)
Apostol, T.M.: Introduction to Analytic Number Theory. Springer-Verlag, Berlin (1989)
Appel, K., Haken, W.: Every planar map is four colorable: Part I-discharging. Illinois J. Math. 21, 429–490 (1977)
Appel, K., Haken, W.: The solution of the four-color-map problem, Sci. Amer. 237(4) 108–121 (1977)
Appel, K., Haken, W., Koch, J.: Every planar map is four colorable: Part II—reducibility. Illinois J. Math. 21, 491–567 (1977)
Aravamudhan, R., Rajendran, B.: Personal communication
Balakrishnan, R.: The energy of a graph. Linear Algebra Appl. 387, 287–295 (2004)
Balakrishnan, R., Paulraja, P.: Powers of chordal graphs. J. Austral. Math. Soc. Ser. A 35, 211–217 (1983)
Balakrishnan, R., Paulraja, P.: Line graphs of subdivision graphs. J. Combin. Info. Sys. Sci. 10, 33–35 (1985)
Balakrishnan, R., Paulraja, P.: Chordal graphs and some of their derived graphs. Congressus Numerantium 53, 71–74 (1986)
Balakrishnan, R., Kavaskar, T., So, W.: The energy of the Mycielskian of a regular graph. Australasian J. Combin. to appear
Barcalkin, A.M., German, L.F.: The external stability number of the Cartesian product of graphs. Bul. Akad. Štiince RSS Moldoven 94(1), 5–8 (1979)
Behzad, M., Chartrand, G., Lesniak-Foster, L.: Graphs and Digraphs. Prindle, Weber & Schmidt International Series, Boston, MA (1979)
Beineke, L.W.: On derived graphs and digraphs. In: Sachs, H., Voss, H.J., Walther, H. (eds.) Beiträge zur Graphentheorie, pp. 17–33. Teubner, Leipzig (1968)
Benzer, S.: On the topology of the genetic fine structure. Proc. Nat. Acad. Sci. USA 45, 1607–1620 (1959)
Berge, C.: Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind. Wiss Z Martin-Luther-Univ. Halle-Wittenberg Math. Natur. Reihe 10, 114–115 (1961)
Berge, C.: Graphs and Hypergraphs. North-Holland, London (1973)
Berge, C.: Graphs and Hypergraphs. North-Holland, Amsterdam (1973)
Berge, C.: Graphs. Elsevier Science, North-Holland, London (1991)
Berge, C., Chvátal, V.: Topics on perfect graphs. Annals of Discrete Mathematics, vol. 21. North Holland, Amsterdam (1984)
Biggs, N.: Algebraic Graph Theory, 2nd ed. Cambridge University Press, Cambridge (1993)
Birkhoff, G., Lewis, D.: Chromatic polynomials. Trans. Amer. Math. Soc. 60, 355–451 (1946)
Bondy, J.A.: Pancyclic graphs. J. Combin. Theory Ser. B 11, 80–84 (1971)
Bondy, J.A., Chvátal, V.: A method in graph theory. Discrete Math. 15, 111–135 (1976)
Bondy, J.A., Halberstam, F.Y.: Parity theorems for paths and cycles in graphs. J. Graph Theory 10, 107–115 (1986)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. MacMillan, India (1976)
Bollobás, B.: Extremal Graph Theory. Academic Press, London (1978)
Brešar, B., Rall, D.F.: Fair reception and Vizing’s conjecture. J. Graph Theory 61(1), 45–54 (2009)
Brešar, B., Dorbec, P., Goddard, W., Hartnell, B.L., Henning, M.A., Klavžar, S., Rall, D.F.: Vizing’s conjecture: A survey and recent results. J. Graph Theory (2011) http://dx.doi.org/10.1002/jgt.20565
Brooks, R.L.: On coloring the nodes of a network. Proc. Cambridge Philos. Soc. 37, 194–197 (1941)
Bryant, D.: Another quick proof that K 10≠P + P + P. Bull. ICA Appl. 34, 86 (2002)
Cayley, A.: On the theory of analytical forms called trees. Philos. Mag. 13, 172–176 (1857); Mathematical Papers, Cambridge 3, 242–246 (1891)
Chartrand, G., Ollermann, O.R.: Applied and algorithmic graph theory. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1993)
Chartrand, G., Wall, C.E.: On the Hamiltonian index of a graph. Studia Sci. Math. Hungar. 8, 43–48 (1973)
Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)
Chung, F.R.K.: Diameters and eigenvalues. J. Amer. Math. Soc. 2, 187–196 (1989)
Chvátal, V.: On Hamilton’s ideals. J. Combin. Theory Ser. B 12, 163–168 (1972)
Chvátal, V., Erdős, P.: A note on Hamiltonian circuits. Discrete Math. 2, 111–113 (1972)
Chvátal, V., Sbihi, N.: Bull-free Berge graphs are perfect. Graphs Combin. 3, 127–139 (1987)
Clark, J., Holton, D.A.: A First Look at Graph Theory. World Scientific, Teaneck, NJ (1991)
Clark, W.E., Suen, S.: Inequality related to Vizing’s conjecture. Electr. J. Combinator. 7 (2000)
Clark, W.E., Ismail, M.E.H., Suen, S.: Application of upper and lower bounds for the domination number to Vizing’s conjecture. Ars Combin. 69, 97–108 (2003)
Cockayne, E.J.: Domination of undirected graphs—a survey, In: Alavi, Y., Lick, D.R. (eds.) Theory and Application of Graphs in America’s Bicentennial Year. Springer-Verlag, New York (1978)
Cockayne, E.J., Hedetniemi, S.T., Miller, D.J.: Properties of hereditary hypergraphs and middle graphs. Networks 7, 247–261 (1977)
Cvetković, D.M.: Graphs and their spectra. Thesis, Univ. Beograd Publ. Elektrotehn. Fak., Ser. Mat. Fiz., No. 354–356, pp. 01–50 (1971)
Cvetković, D.M., Doob, M., Sachs, H.: Spectra of Graphs—Theory and Application, Third revised and enlarged edition. Johann Ambrosius Barth Verlag, Heidelberg/Leipzig (1995)
Deligne, P.: La conjecture de Weil I. Publ. Math. IHES 43, 273–307 (1974)
Demoucron, G., Malgrange, Y., Pertuiset, R.: Graphes planaires: Reconnaissance et construction de représentaitons planaires topologiques. Rev. Française Recherche Opérationnelle 8, 33–47 (1964)
Deo, N.: Graph Theory with Applications to Engineering and Computer Science. Prentice-Hall, Englewood Cliffs, NJ (1974)
Descartes, B.: Solution to advanced problem no. 4526. Am. Math. Mon. 61, 269–271 (1954)
Diestel, R.: Graph theory. In: Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010)
Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Math. 1, 269–271 (1959)
Dirac, G.A.: A property of 4-chromatic graphs and some remarks on critical graphs. J. London Math. Soc. 27, 85–92 (1952)
Dirac, G.A.: Some theorems on abstract graphs. Proc. London Math. Soc. 2, 69–81 (1952)
Dirac, G.A.: 4-chrome Graphen und vollständige 4-Graphen. Math. Nachr. 22, 51–60 (1960)
Dirac, G.A.: Généralisations du théoréme de Menger. C.R. Acad. Sci. Paris 250, 4252–4253 (1960)
Dirac, G.A.: On rigid circuit graphs-cut sets-coloring. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)
Drinfeld, V.: The proof of Peterson’s conjecture for GL(2) over a global field of characteristic p. Funct. Anal. Appl. 22, 28–43 (1988)
Droll, A.: A classification of Ramanujan Cayley graphs. Electron. J. Cominator. 17, #N29 (2010)
El-Zahar, M., Pareek, C.M.: Domination number of products of graphs. Ars Combin. 31 (1991) 223–227
Fáry, I.: On straight line representation of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)
Ferrar, W.L.: A Text-Book of Determinants, Matrices and Algebraic Forms. Oxford University Press (1953)
Fiorini, S., Wilson, R.J.: Edge-colourings of graphs. Research Notes in Mathematics, vol. 16. Pitman, London (1971)
Fleischner, H.: Elementary proofs of (relatively) recent characterizations of Eulerian graphs. Discrete Appl. Math. 24, 115–119 (1989)
Fleischner, H.: Eulerian graphs and realted topics. Ann. Disc. Math. 45 (1990)
Ford, L.R. Jr., Fulkerson, D.R.: Maximal flow through a network. Canad. J. Math. 8, 399–404 (1956)
Ford, L.R. Jr., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)
Fournier, J.-C.: Colorations des arétes d’un graphe. Cahiers du CERO 15, 311–314 (1973)
Fournier, J.-C.: Demonstration simple du theoreme de Kuratowski et de sa forme duale. Discrete Math. 31, 329–332 (1980)
Fulkerson, D.R.: Blocking and anti-blocking pairs of polyhedra. Math. Programming 1, 168–194 (1971)
Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pacific J. Math. 15, 835–855 (1965)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman & Co., San Francisco (1979)
Gibbons, A.: Algorithmic Graph Theory. Cambridge University Press, Cambridge (1985)
Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and interval graphs. Canad. J. Math. 16, 539–548 (1964)
Goddard, W.D., Kubicki, G., Ollermann, O.R., Tian, S.L.: On multipartite tournaments. J. Combin. Theory Ser. B 52, 284–300 (1991)
Godsil, C.D., Royle, G.: Algebraic graph theory. Graduate Texts in Mathematics, vol. 207. Springer-Verlag, Berlin (2001)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)
Gould, R.J.: Graph Theory. Benjamin/Cummings Publishing Company, Menlo Park, CA (1988)
Grinberg, É.Ja.: Plane homogeneous graphs of degree three without Hamiltonian circuits (Russian). Latvian Math. Yearbook 4, 51–58 (1968)
Gross, J.L., Tucker, T.W.: Topological Graph Theory. Wiley, New York (1987)
Gross, J.L., Yellen, J.: Handbook of graph theory. Discrete Mathematics and Its Applications, vol. 25. CRC Press, Boca Raton, FL (2004)
Gupta, R.P.: The chromatic index and the degree of a graph. Notices Amer. Math. Soc. 13, Abstract 66T–429 (1966)
Gutman, I.: The energy of a graph. Ber. Math. Stat. Sekt. Forschungszent. Graz. 103, 1–22 (1978)
Gutman, I.: The energy of a graph: Old and new results. In: Betten, A., Kohnert, A., Laue, R., Wassermann, A. (eds.) Algebraic Combinatorics and Applications, pp. 196–211, Springer, Berlin (2001)
Gutman, I., Polansky, O.: Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin (1986)
Gutman, I., Zhou, B.: Laplacian energy of a graph. Linear Algebra Appl. 414, 29–37 (2006)
Gutman, I., Soldatović, T., Vidović, D.: The energy of a graph and its size dependence. A Monte Carlo approach. Chem. Phys. Lett. 297, 428–432 (1998)
Gutman, I., Firoozabadi, S.Z., de la Pẽna, J.A., Rada, J.: On the energy of regular graphs. MATCH Commun. Math. Comput. Chem. 57, 435–442 (2007)
Gyárfás, A., Lehel, J., Nešetril, J., Rödl, V., Schelp, R.H., Tuza, Z.: Local k-colorings of graphs and hypergraphs. J. Combin. Theory Ser. B 43, 127–139 (1987)
Hajnal, A., Surányi, J.: Über die Auflösung von Graphen in Vollständige Teilgraphen. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 1, 113–121 (1958)
Hall, P.: On representatives of subsets. J. London Math. Soc. 10, 26–30 (1935)
Hall, M. Jr.: Combinatorial Theory. Blaisdell, Waltham, MA (1967)
Harary, F.: The determinant of the adjacency matrix of a graph. SIAM Rev 4, 202–210 (1962)
Harary, F.: Graph Theory. Addison-Wesley, Reading, MA (1969)
Harary, F., Nash-Williams, C.St.J.A.: On Eulerian and Hamiltonian graphs and line graphs. Canad. Math. Bull. 8, 701–710 (1965)
Harary, F., Palmer, E.M.: Graphical Enumeration, Academic Press, New York (1973)
Harary, F., Tutte, W.T.: A dual form of Kuratowski’s theorem. Canad. Math. Bull. 8, 17–20 (1965)
Harary, F., Norman, R.Z., Cartwright, D.: Structural Models: An Introduction to the Theory of Directed Graphs. Wiley, New York (1965)
Hartnell, B., Rall, D.F.: On Vizing’s conjecture. Congr. Numer. 82, 87–96 (1991)
Hartnell, B., Rall, D.F.: Domination in Cartesian products: Vizing’s conjecture. In: Domination in Graphs, Advanced Topics, vol. 209. Monographs and Textbooks in Pure and Applied Mathematics, pp. 163–189. Marcel Dekker, New York (1998)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)
Hayward, R.B.: Weakly triangulated graphs. J. Combin. Theory Ser. B 39, 200–208 (1985)
Heawood, P.J.: Map colour theorems. Quart. J. Math. 24, 332–338 (1890)
Hedetniemi, S.T.: Homomorphisms of graphs and automata. Tech. Report 03105-44-T, University of Michigan (1966)
Hell, P., Nešetril, J.: Graphs and homorphisms. Oxford Lecture Series in Mathematics and its Applications, vol. 28. Oxford University Press, Oxford (2004)
Holton, D.A., Sheehan, J.: The Petersen graph. Australian Mathematical Society Lecture Series, vol. 7, Cambridge University Press, Cambridge (1993)
Hougardy, S., Le, V.B., Wagler, A.: Wing-triangulated graphs are perfect. J. Graph Theory 24, 25–31 (1997)
Ilić, A.: The energy of unitary Cayley graphs. Linear Algebra Appl. 431, 1881–1889 (2009)
Ilić, A.: Distance spectra and distance energy of integral circulant graphs. Linear Algebra Appl. 433, 1005–1014 (2010)
Indulal, G., Gutman, I., Vijayakumar, A.: On distance energy of graphs. MATCH Commun. Math. Comput. Chem. 60, 461–472 (2008)
Irving, R.W., Manlove, D.F.: The b-chromatic number of a graph. Discrete Appl. Math. 91, 127–141 (1999)
Jacobson, M.S., Kinch, L.F.: On the domination number of product graphs: I. Ars. Combin. 18, 33–44 (1984)
Jaeger, F.: A note on sub-Eulerian graphs. J. Graph Theory 3, 91–93 (1979)
Jaeger, F.: Nowhere-zero flow problems. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory III, pp. 71–95. Academic Press, London (1988)
Jaeger, F., Payan, C.: Relations du type Nordhaus–Gaddum pour le nombre d’absorption d’un graphe simple. C. R. Acad. Sci. Paris A 274, 728–730 (1972)
Jensen, T.R., Toft, B.: Graph coloring problems. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1995)
Jordan, C.: Sur les assemblages de lignes. J. Reine Agnew. Math. 70, 185–190 (1869)
Jüng, H.: Zu einem Isomorphiesatz von Whitney für Graphen. Math. Ann. 164, 270–271 (1966)
Jünger, M., Pulleyblank, W.R., Reinelt, G.: On partitioning the edges of graphs into connected subgraphs. J. Graph Theory 9, 539–549 (1985)
Kainen, P.C., Saaty, T.L.: The Four-Color Problem (Assaults and Conquest). Dover Publications, New York (1977)
Kempe, A.: On the geographical problem of four colours. Amer. J. Math. 2, 193–200 (1879)
Kilpatrick, P.A.: Tutte’s first colour-cycle conjecture. Ph.D. thesis, Cape Town, (1975)
Klotz, W., Sander, T.: Some properties of unitary Cayley graphs. Electron. J. Combinator. 14, 1–12 (2007)
Koolen, J.H., Moulton, V.: Maximal energy graphs. Adv. Appl. Math. 26, 47–52 (2001)
Kouider, M., Mahéo, M.: Some bounds for the b-chromatic number of a graph. Discrete Math. 256, 267–277 (2002)
Kratochvíl, J., Tuza, Z., Voigt, M.: On the b-chromatic number of graphs. Lecture Notes Comput. Sci. 2573, 310–320 (2002)
Kruskal, J.B. Jr.: On the shortest spanning subtree of a graph and the travelling salesman problem. Proc. Amer. Math. Soc. 7, 48–50 (1956)
Kundu, S.: Bounds on the number of disjoint spanning trees. J. Combin. Theory Ser. B 17, 199–203 (1974)
Kuratowski, C.: Sur le problème des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)
Laskar, R., Shier, D.: On powers and centers of chordal graphs. Discrete Applied Math. 6, 139–147 (1983)
Lesniak, L.M.: Neighborhood unions and graphical properties. In: Alavi, Y., Chartrand, G., Ollermann, O.R., Schwenk, A.J. (eds.) Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs: Graph Theory, Combinatorics and Applications. Western Michigan University, pp. 783–800. Wiley, New York (1991)
Li, W.C.W.: Number theory with applications. Series of University Mathematics, vol. 7. World Scientific, Singapore (1996)
Li, X., Li, Y., Shi, Y.: Note on the energy of regular graphs. Linear Algebra Appl. 432, 1144–1146 (2010)
Lovász, L.: Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2, 253–267 (1972)
Lovász, L.: Three short proofs in graph theory. J. Combin Theory Ser. B 19, 111–113 (1975)
Lovász, L., Plummer, M.D.: Matching theory. Annals of Discrete Mathematics, vol. 29. North-Holland Mathematical Studies, vol. 121 (1986)
Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8, 261–277 (1988)
McKee, T.A.: Recharacterizing Eulerian: Intimations of new duality. Discrete Math. 51, 237–242 (1984)
Meir, A., Moon, J.W.: Relations between packing and covering numbers of a tree. Pacific J. Math. 61, 225–233 (1975)
Menger, K.: Zur allgemeinen Kurventheorie. Fund. Math. 10, 96–115 (1927)
Moon, J.W.: On subtournaments of a tournament. Canad. Math. Bull. 9, 297–301 (1966)
Moon, J.W.: Various proofs of Cayley’s formula for counting trees. In: Harary, F. (eds.) A Seminar on Graph Theory, pp. 70–78. Holt, Rinehart and Winston, Inc., New York (1967)
Moon, J.W.: Topics on Tournaments. Holt, Rinehart and Winston Inc., New York (1968)
Mycielski, J.: Sur le coloriage des graphs. Colloq. Math. 3, 161–162 (1955)
Nash-Williams, C.St.J.A.: Edge-disjoint spanning trees of finite graphs. J. London Math. Soc. 36, 445–450 (1961)
Nebesky, L.: On the line graph of the square and the square of the line graph of a connected graph, Casopis. Pset. Mat. 98, 285–287 (1973)
Nikiforov, V.: The energy of graphs and matrices. J. Math. Anal. Appl. 326, 1472–1475 (2007)
Nordhaus, E.A., Gaddum, J.W.: On complementary graphs. Amer. Math. Monthly 63, 175–177 (1956)
Oberly, D.J., Sumner, D.P.: Every connected, locally connected nontrivial graph with no induced claw is Hamiltonian. J. Graph Theory 3, 351–356 (1979)
Ore, O.: Note on Hamilton circuits. Amer. Math. Monthly 67, 55 (1960)
Ore, O.: Theory of graphs. Amer. Math. Soc. Transl. 38, 206–212 (1962)
Ore, O.: The Four-Color Problem. Academic Press, New York (1967)
Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Upper Saddle River, NJ (1982)
Parthasarathy, K.R., Ravindra, G.: The strong perfect-graph conjecture is true for K 1, 3-free graphs. J. Combin. Theory Ser. B 21, 212–223 (1976)
Parthasarathy, K.R.: Basic Graph Theory. Tata McGraw-Hill Publishing Company Limited, New Delhi (1994)
Parthasarathy, K.R., Ravindra, G.: The validity of the strong perfect-graph conjecture for (K 4 − e)-free graphs. J. Combin. Theory Ser. B 26, 98–100 (1979)
Peña, I., Rada, J.: Energy of digraphs. Linear and Multilinear Algebra 56(5), 565–579 (2008)
Petersen, J.: Die Theorie der regulären Graphen. Acta Math. 15, 193–220 (1891)
Prim, R.C.: Shortest connection networks and some generalizations. Bell System Techn. J. 36, 1389–1401 (1957)
Rall, D.F.: Total domination in categorical products of graphs. Discussiones Mathematicae Graph Theory 25, 35–44 (2005)
Ramaswamy, H.N., Veena, C.R.: On the energy of unitary Cayley graphs. Electron. J. Combinator. 16 (2009)
Ram Murty, M.: Ramanujan graphs. J. Ramanujan Math. Soc. 18(1), 1–20 (2003)
Ram Murty, M.: Ramanujan graphs and zeta functions. Jeffery-Williams Prize Lecture. Canadian Mathematical Society, Canada (2003)
Ray-Chaudhuri, D.K., Wilson, R.J.: Solution of Kirkman’s schoolgirl problem. In: Proceedings of the Symposium on Mathematics, vol. 19, pp. 187–203. American Mathematical Society, Providence, RI (1971)
Rédei, L.: Ein kombinatorischer satz. Acta. Litt. Sci. Szeged 7, 39–43 (1934)
Roberts, F.S.: Graph theory and its applications to problems in society. CBMS-NSF Regional Conference Series in Mathematics. SIAM, Philadelphia (1978)
Sachs, H.: Über teiler, faktoren und charakteristische polynome von graphen II. Wiss. Z. Techn. Hochsch. Ilmenau 13, 405–412 (1967)
Sampathkumar, E.: A characterization of trees. J. Karnatak Univ. Sci. 32, 192–193 (1987)
Schwenk, A.J., Lossers, O.P.: Solutions of advanced problems. Am. Math. Mon. 94, 885–887 (1987)
Serre, J.-P.: Trees. Springer-Verlag, New York (1980)
Shader, B., So, W.: Skew spectra of oriented graphs. Electron. J. Combinator. 16, 1–6 (2009)
Shrikhande, S.S., Bhagwandas: Duals of incomplete block designs. J. Indian Stat. Assoc. 3, 30–37 (1965)
Stevanović, D., Stanković, I.: Remarks on hyperenergetic circulant graphs. Linear Algebra Appl. 400, 345–348 (2005)
Sumner, D.P.: Graphs with 1-factors. Proc. Amer. Math. Soc. 42, 8–12 (1974)
Toida, S.: Properties of an Euler graph. J. Franklin. Inst. 295, 343–346 (1973)
Trinajstic, N.: Chemical Graph Theory—Volume I. CRC Press, Boca Raton, FL (1983)
Trinajstic, N.: Chemical Graph Theory—Volume II. CRC Press, Boca Raton, FL (1983)
Tucker, A.: The validity of perfect graph conjecture for K 4-free graphs. In: Berge, C., Chvátal, V. (eds.) Topics on Perfect Graphs, vol. 21, pp. 149–157 (1984)
Tutte, W.T.: The factorization of linear graphs. J. London Math. Soc. 22, 107–111 (1947)
Tutte, W.T.: A theorem on planar graphs. Trans. Amer. Math. Soc. 82, 570–590 (1956)
Tutte, W.T.: On the problem of decomposing a graph into n connected factors. J. London Math. Soc. 36, 221–230 (1961)
Vizing, V.G.: The Cartesian product of graphs. Vycisl/Sistemy 9, 30–43 (1963)
Vizing, V.G.: On an estimate of the chromatic class of a p-graph (in Russian). Diskret. Analiz. 3, 25–30 (1964)
Vizing, V.G.: A bound on the external stability number of a graph. Dokl. Akad. Nauk. SSSR 164, 729–731 (1965)
Vizing, V.G.: Some unsolved problems in graph theory. Uspekhhi Mat. Nauk. 23(6), 117–134 (1968)
Wagner, K.: Über eine eigenschaft der ebenen komplexe. Math. Ann. 114, 570–590 (1937)
Walikar, H.B., Acharya, B.D., Sampathkumar, E.: Recent developments in the theory of domination in graphs. MRI Lecture Notes in Mathematics, vol. 1. Mehta Research Institue, Allahabad (1979)
Walikar, H.B., Ramane, H.S., Hampiholi, P.R.: On the energy of a graph. In: Mulder, H.M., Vijayakumar, A., Balakrishnan, R. (eds.) Graph Connections, pp. 120–123. Allied Publishers, New Delhi (1999)
Walikar, H.B., Gutman, I., Hampiholi, P.R., Ramane, H.S.: Non-hyperenergetic graphs. Graph Theory Notes New York 41, 14–16 (2001)
Walikar, H.B., Ramane, H.S., Jog, S.R.: On an open problem of R. Balakrishnan and the energy of products of graphs. Graph Theory Notes New York 55, 41–44 (2008)
Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)
West, D.B.: Introduction to Graph Theory, 2nd ed. Prentice Hall, New Jersey (2001)
Whitney, H.: Congruent graphs and the connectivity of graphs. Amer. J. Math. 54, 150–168 (1932)
Yap, H.P.: Some topics in graph theory. London Mathematical Society Lecture Notes Series, vol. 108, Cambridge University Press, Cambridge (1986)
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Balakrishnan, R., Ranganathan, K. (2012). Domination in Graphs. In: A Textbook of Graph Theory. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4529-6_10
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