Question

Prove that if G is a connected graph of order n ≥ 2, then the vertices of G can be listed as v₁, v₂, . . . , v_n such that each vertex v_i (2 ≤ i ≤ n) is adjacent to some vertex in the set {v₁, v₂, . . . , v_i−1}.

Answer 1

Solution:

nG is k-coppested arah k2 an are t+1 distinct vertices th o prove hat Gr contains a u-vipa .poY eachi Kiet, and two , which have only uincommon we will prove above Statmen indu chon on ster 1 : Let k=2 e. Gris2-connected 9 rap il since it is 2-connected graph it will have minimum three eatges It clearly contain u-u & u-½ path of which only u is common sfatment is nt is Utrue for k 2 sip2: Assume statment statment for k-n-I rove statment oy k=n n-conn ected graph and e. G is lu raph an verbices of G . le 曳 n-l

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By ineluction hypothe L s of which have onh as only one Consider u,,l, verterx u (for undersandi So G7、s divided into hoa paHs Nou G is conpected. Hence the e 足、 erísk an edge between u artherGis n-connected By using above edge pa get U- we Hence we gd u paths Isn which have only u in common By induchon above corollary strue

Answer 2

Solution:

nG is k-coppested arah k2 an are t+1 distinct vertices th o prove hat Gr contains a u-vipa .poY eachi Kiet, and two , which have only uincommon we will prove above Statmen indu chon on ster 1 : Let k=2 e. Gris2-connected 9 rap il since it is 2-connected graph it will have minimum three eatges It clearly contain u-u & u-½ path of which only u is common sfatment is nt is Utrue for k 2 sip2: Assume statment statment for k-n-I rove statment oy k=n n-conn ected graph and e. G is lu raph an verbices of G . le 曳 n-l

*********************************************************

By ineluction hypothe L s of which have onh as only one Consider u,,l, verterx u (for undersandi So G7、s divided into hoa paHs Nou G is conpected. Hence the e 足、 erísk an edge between u artherGis n-connected By using above edge pa get U- we Hence we gd u paths Isn which have only u in common By induchon above corollary strue