For a family of graphs
n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n>, a
n n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n>-tra
nsversal of a graph
n id="mmlsi27" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si27.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=487d4e16b1091219267cc1dd33f354d4" title="Click to view the MathML source">Gn>n class="mathContainer hidden">n class="mathCode">n>n>n> is a subset
n id="mmlsi139" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si139.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=9ab7640b0c3c14009bec35735c808739" title="Click to view the MathML source">S⊆V(G)n>n class="mathContainer hidden">n class="mathCode">n>n>n> that i
ntersects every subset of
n id="mmlsi140" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si140.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=ea1c4dbd06cc7b7be8ea304b4df4ae63" title="Click to view the MathML source">V(G)n>n class="mathContainer hidden">n class="mathCode">n>n>n> that i
nduces a subgraph isomorphic to a graph i
n n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n>. Let
n id="mmlsi142" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si142.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=19b12ebb72bcc2d52baee1004af9ea64" title="Click to view the MathML source">tF(G)n>n class="mathContainer hidden">n class="mathCode">n>n>n> be the mi
nimum size of a
n n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n>-tra
nsversal of
n id="mmlsi27" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si27.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=487d4e16b1091219267cc1dd33f354d4" title="Click to view the MathML source">Gn>n class="mathContainer hidden">n class="mathCode">n>n>n>, a
nd
n id="mmlsi145" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si145.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=ae7b0a86adb0226bcdeb601bfff08710">nlineImage" height="15" width="46" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0195669816300439-si145.gif"><noscript>n:bottom" width="46" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0195669816300439-si145.gif">noscript>n class="mathContainer hidden">n class="mathCode">n>n>n> be the mi
nimum size of a
n n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n>-tra
nsversal of
n id="mmlsi27" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si27.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=487d4e16b1091219267cc1dd33f354d4" title="Click to view the MathML source">Gn>n class="mathContainer hidden">n class="mathCode">n>n>n> that i
nduces a co
nnected graph. For a class of co
nnected graphs
n id="mmlsi148" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si148.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=144219da194008c5ccf835ff5f9ce40b" title="Click to view the MathML source">Gn>n class="mathContainer hidden">n class="mathCode">n>n>n>, we say that the
price of co
nnectivity of
n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n>-tra
nsversals is multiplicative if, for all
n id="mmlsi150" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si150.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=20b0dba435b879ff24ffeef1de663543" title="Click to view the MathML source">G∈Gn>n class="mathContainer hidden">n class="mathCode">n>n>n>,
n id="mmlsi151" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si151.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=abea3767f31fb2b2ac5f3c7996634452">nlineImage" height="15" width="91" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0195669816300439-si151.gif"><noscript>n:bottom" width="91" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0195669816300439-si151.gif">noscript>n class="mathContainer hidden">n class="mathCode">n>n>n> is bou
nded by a co
nsta
nt, a
nd additive if
n id="mmlsi152" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si152.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=4fd12abe4223081764b990081b68ec59">nlineImage" height="15" width="104" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0195669816300439-si152.gif"><noscript>n:bottom" width="104" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0195669816300439-si152.gif">noscript>n class="mathContainer hidden">n class="mathCode">n>n>n> is bou
nded by a co
nsta
nt. The
price of co
nnectivity is ide
ntical if
n id="mmlsi142" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si142.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=19b12ebb72bcc2d52baee1004af9ea64" title="Click to view the MathML source">tF(G)n>n class="mathContainer hidden">n class="mathCode">n>n>n> a
nd
n id="mmlsi145" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si145.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=ae7b0a86adb0226bcdeb601bfff08710">nlineImage" height="15" width="46" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0195669816300439-si145.gif"><noscript>n:bottom" width="46" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0195669816300439-si145.gif">noscript>n class="mathContainer hidden">n class="mathCode">n>n>n> are always equal a
nd u
nbou
nded if
n id="mmlsi145" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si145.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=ae7b0a86adb0226bcdeb601bfff08710">nlineImage" height="15" width="46" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0195669816300439-si145.gif"><noscript>n:bottom" width="46" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0195669816300439-si145.gif">noscript>n class="mathContainer hidden">n class="mathCode">n>n>n> ca
nnot be bou
nded i
n terms of
n id="mmlsi142" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si142.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=19b12ebb72bcc2d52baee1004af9ea64" title="Click to view the MathML source">tF(G)n>n class="mathContainer hidden">n class="mathCode">n>n>n>. We study classes of graphs characterized by o
ne forbidde
n i
nduced subgraph
n id="mmlsi9" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si9.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=120b43f1c34315c12cedc7aa06db6287" title="Click to view the MathML source">Hn>n class="mathContainer hidden">n class="mathCode">n>n>n> a
nd
n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n>-tra
nsversals where
n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n> co
ntai
ns a
n i
nfi
nite
number of cycles a
nd, possibly, also o
ne or more a
nticycles or short paths. We determi
ne exactly those classes of co
nnected
n id="mmlsi9" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si9.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=120b43f1c34315c12cedc7aa06db6287" title="Click to view the MathML source">Hn>n class="mathContainer hidden">n class="mathCode">n>n>n>-free graphs where the
price of co
nnectivity of these
n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n>-tra
nsversals is u
nbou
nded, multiplicative, additive, or ide
ntical. I
n particular, our tetrachotomies exte
nd k
now
n results for the case whe
n n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300439&_mathId=si4.gif&_user=111111111&_pii=S0195669816300439&_rdoc=1&_issn=01956698&md5=c686c709b0820f48160c7af18565ae55" title="Click to view the MathML source">Fn>n class="mathContainer hidden">n class="mathCode">n>n>n> is the family of all cycles.