<PAGE>   1
                                                                   Exhibit 10.13




                             STOCKHOLDER'S AGREEMENT
                           DATED AS OF JUNE ___, 2000






                             STOCKHOLDER'S AGREEMENT



                            DATED AS OF JUNE __, 2000


                                     BETWEEN


                             UCAR INTERNATIONAL INC.

                                       AND

                                  GRAFTECH INC.
<PAGE>   2
                                TABLE OF CONTENTS

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                                                                                        PAGE

<S>                                                                                     <C>
section 1.    DEFINITIONS                                                                  1

         1.1      Defined Terms                                                            1

         1.2      Other Definitional Provisions; Interpretation                            2

section 2.    TRANSFERS                                                                    3

         2.1      Limitations on Transfer                                                  3

         2.2      Transfers to Affiliates and Others                                       3

         2.3      Effect of Void Transfers                                                 3

         2.4      Legends                                                                  3

section 3    PREEMPTIVE RIGHTS                                                             4

         3.1      General                                                                  4

         3.2      Exercise                                                                 4

section 4     REGISTRATION RIGHTS                                                          4

         4.1      Demand Registration                                                      4

         4.2      Piggy-Back Rights                                                        5

         4.3      Other Registration-Related Matters                                       6

         4.4      Indemnification                                                          6

section 5    MANAGEMENT                                                                    9

         5.1      Composition of Board                                                     9

section 6    ACTIONS REQUIRING UCAR APPROVAL                                              10

         6.1      Specified Actions                                                       10

section 7    OTHER AGREEMENTS                                                             11

         7.1      Non-Solicitation                                                        11

         7.2      Financial Information                                                   11

         7.3      Confidentiality                                                         12

         7.4      Initial Public Offering Expenses                                        13

         7.5      Third Party Beneficiary Provisions; Listing and Registration            13

section 8    MISCELLANEOUS                                                                13

         8.1      Additional Securities Subject to Agreement                              13

         8.2      Injunctive Relief                                                       14

         8.3      Amendments                                                              14
</TABLE>

                                      -i-
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                                TABLE OF CONTENTS
                                  (CONTINUED)
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                                                                                         PAGE
<S>                                                                                     <C>
         8.4      Successors, Assigns and Transferees                                     14

         8.5      Notices                                                                 14

         8.6      Integration                                                             15

         8.7      Severability                                                            15

         8.8      Governing Law                                                           15

         8.9      Counterparts                                                            15
</TABLE>

                                      -ii-
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         STOCKHOLDER'S AGREEMENT, dated as of ________, 2000, between UCAR
International Inc., a Delaware corporation ("UCAR International"), and Graftech
Inc., a Delaware corporation (the "Company").

                              W I T N E S S E T H :

         WHEREAS, one of the wholly-owned Subsidiaries (as defined below) of
UCAR International owns an aggregate of ____________ shares of common stock, par
value $.01 per share ("Common Stock"), of the Company and is the sole
stockholder of the Company; and

         WHEREAS, UCAR International and the Company propose an initial public
offering of outstanding and newly issued shares of Common Stock (the "IPO")
following which UCAR International will directly or indirectly own not less than
80% of the then outstanding shares of Common Stock; and

         WHEREAS, the parties wish to provide for the structure of the Board (as
defined below) and certain matters related to the operations of the Company, to
provide for the sale of shares of Common Stock, to make other provisions with
respect to the affairs of the Company and to provide UCAR International and its
Subsidiaries other than the Company and its Subsidiaries (collectively, "UCAR")
with certain rights, including registration rights; and

         WHEREAS, the Company and UCAR have agreed to enter into this Agreement
to accomplish the foregoing.

         NOW, THEREFORE, in consideration of the mutual covenants and agreements
herein contained, the parties agree as follows:

SECTION 1.        DEFINITIONS

         1.1      Defined Terms. As used in this Agreement, the following
capitalized terms shall have the meanings ascribed to them below:

                  "Affiliate" shall mean, with respect to any Person, (i) any
Person that directly or indirectly controls, is controlled by or is under common
control with such Person or (ii) any director, officer, general partner or
employee of such Person or any Person specified in clause (i) above; provided,
that (x) UCAR, on the one hand, and the Company and its Subsidiaries, on the
other hand, shall not be deemed to be Affiliates of each other for purposes
hereof and (y) officers, directors or employees of the Company or its
Subsidiaries shall not be deemed to be Affiliates of UCAR for purposes hereof
solely by reason of being officers, directors or employees of the Company or its
Subsidiaries.

                  "Agreement" shall mean this Stockholder's Agreement, as the
same may be amended, supplemented or otherwise modified from time to time.

                  "Board" shall mean the Board of Directors of the Company.

<PAGE>   5
                   "Graftech Business" shall mean the development, manufacture
and sale of natural, acid-treated and flexible graphite-based products.

                   "Person" shall mean any individual or any corporation,
limited liability company, partnership, trust, joint stock company, business
trust, unincorporated association, joint venture, governmental authority or
other entity of any nature whatsoever.

                  "Public Offering" shall mean the sale of shares of Common
Stock to the public pursuant to an effective registration statement filed under
the Securities Act, including the IPO.

                  "Registration Expenses" shall mean any and all expenses
incident to performance of or compliance with Section 4.1, 4.2 or 4.3,
including, without limitation, (i) all SEC and all stock exchange, stock market
and self-regulatory organization registration and filing fees, (ii) all fees and
expenses of complying with foreign or state securities or blue sky laws
(including fees and disbursements of counsel for the underwriters in connection
with such compliance), (iii) all printing and related messenger and delivery
expenses, (iv) the fees and disbursements of counsel for the Company and of its
independent public accountants, including the expenses of any special audits
and/or "cold comfort" letters required by or incident to such performance and
compliance, (v) the reasonable fees and disbursements of one counsel, other than
the Company's counsel, selected by UCAR, to represent it in connection with each
registration under any of such Sections, and (vi) any fees and disbursements of
underwriters customarily paid by the issuers or sellers of securities, and the
reasonable fees and expenses of any special due diligence or other investigators
or experts retained in connection with such registration, but excluding
underwriting discounts and commissions and transfer, stamp or similar taxes, if
any.

                  "SEC" shall mean the Securities and Exchange Commission.

                  "Securities Act" shall mean the Securities Act of 1933, as
amended, and the rules and regulations promulgated thereunder, as the same may
be amended from time to time.

                  "Subsidiary" of a Person shall mean any entity of which such
Person owns, directly or indirectly or through one or more other Subsidiaries,
stock, ownership interests or other rights or securities having a majority of
the ordinary voting power of such entity.

                  "Third Party" shall mean any Person other than an Affiliate of
UCAR or the Company.

                  "Transfer" shall mean any transfer, gift, sale, assignment,
exchange, mortgage, pledge, hypothecation or other disposition (whether for or
without consideration and whether voluntary or involuntary or by operation of
law).

         1.2      Other Definitional Provisions; Interpretation.

                  (a)      The words "hereof," "herein," "hereunder" and words
of similar import when used in this Agreement shall refer to this Agreement as a
whole and not to any particular provision of this Agreement, and Section and
Subsection references refer to Sections and

                                       2
<PAGE>   6
Subsections of this Agreement unless otherwise specified.

                  (b)      The headings in this Agreement are included for
convenience of reference only and shall not limit or otherwise affect the
meaning or interpretation of this Agreement.

                  (c)      The meanings given to terms defined herein shall be
equally applicable to both the singular and plural forms of such terms.

                  (d)      The term "including" shall mean "including, without
limitation".

                  (e)      Any reference to any agreement, instrument, contract
or other document shall (i) include all appendices, exhibits, annexes and
schedules thereto and (ii) be a reference to such agreement, instrument,
contract or other document as amended, supplemented, modified, suspended,
restated or novated from time to time.

SECTION 2.        TRANSFERS

         2.1      Limitations on Transfer. UCAR agrees that (i) except for
Transfers effected in connection with a Public Offering or pursuant to Section
2.2(a) or (c), it shall not Transfer any shares of Common Stock unless it shall
have furnished to the Company an opinion of counsel to the Company that such
Transfer is exempt from the provisions of Section 5 under the Securities Act and
(ii) except for Transfers in connection with a Public Offering and Transfers
pursuant to Rule 144 under the Securities Act or Section 2.2(d), it shall not
Transfer any shares of Common Stock unless the transferee shall agree to become
a party to, and be bound to the same extent (and entitled to the same benefits)
as its transferor by the terms of, this Agreement.

         2.2      Transfers to Affiliates and Others. UCAR shall be entitled,
from time to time and at any time, to Transfer shares of Common Stock to (a) any
of its Affiliates, (b) any Person who, upon such Transfer, becomes a "Principal
Stockholder" as defined in the Company's Amended and Restated Certificate of
Incorporation as in effect on the date hereof, (c) lenders (or nominees or
agents for lenders) under UCAR's credit facilities or (d) stockholders of UCAR
or a Person described in the preceding clause (b), or the ultimate parent of
such a Person, at a time when UCAR or such Person or parent is a public
reporting company (including a distribution of shares of Common Stock upon a
liquidation thereof.)

         2.3      Effect of Void Transfers. In the event of any purported
Transfer of any shares of Common Stock in violation of the provisions of this
Agreement, such purported Transfer shall be void and the Company shall not give
effect to such Transfer.

         2.4      Legends. The certificates representing shares of Common Stock
held by UCAR shall bear the following legends:

         "TRANSFER OF SHARES REPRESENTED BY THIS CERTIFICATE IS RESTRICTED BY A
         STOCKHOLDER'S AGREEMENT, DATED AS OF ______,

                                       4
<PAGE>   7
         2000, WHICH AGREEMENT CONTAINS ADDITIONAL PROVISIONS AFFECTING THE
         RIGHTS AND DUTIES OF THE HOLDER OF SUCH SHARES. A COPY OF THE
         STOCKHOLDER'S AGREEMENT IS ON FILE AT THE OFFICE OF THE COMPANY."

         "THE SECURITIES REPRESENTED BY THIS CERTIFICATE HAVE NOT BEEN
         REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED, OR UNDER ANY
         STATE SECURITIES LAW. SUCH SECURITIES MAY NOT BE TRANSFERRED, EXCEPT
         PURSUANT TO (I) A REGISTRATION STATEMENT WITH RESPECT TO SUCH
         SECURITIES WHICH IS EFFECTIVE UNDER SUCH ACT OR (II) AN EXEMPTION FROM
         THE REGISTRATION REQUIREMENTS OF SUCH ACT."

Any certificates issued upon a Transfer thereof or in exchange therefor shall,
except as otherwise required to give effect to transactions permitted or
required by this Agreement, bear substantially similar legends. The first legend
referenced above shall be removed from certificates representing all shares of
Common Stock upon termination of this Agreement or from certificates
representing shares of Common Stock when such shares become free and clear of
any restriction or provision hereof. The second legend referenced above shall be
removed from certificates representing shares of Common Stock when such shares
become free and clear of any restriction referenced therein. The Company shall
make appropriate notation in its stock transfer records of the restrictions on
Transfer provided for in this Agreement.

SECTION 3.        PREEMPTIVE RIGHTS

         3.1      General. So long as this Agreement shall remain in effect,
UCAR has the right, upon any issuance by the Company (an "Issuance") of any
additional shares of Common Stock or securities convertible into or exercisable
or exchangeable for shares of Common Stock (other than non-qualified employee or
director stock options which cannot and do not vest prior to the third
anniversary of the effective date of the registration statement filed with the
SEC relating to the IPO (the "Effective Date") or, in the case of non-qualified
director stock options, which cannot and do not vest at a time when UCAR owns or
holds directly or indirectly 20% or more of our then outstanding shares of
Common Stock (collectively, "Permitted Options")), to any Third Party (a
"Recipient"), to receive, 30 days prior to such Issuance to such Recipient,
notification in writing of the proposed Issuance to the proposed Recipient and
has the right (the "Preemptive Right") to subscribe for and purchase additional
shares of Common Stock at the same price and upon the same terms and conditions
as those to be issued in the Issuance to such Recipient such that, immediately
after giving effect to such Issuance to the Recipient and exercise of the
Preemptive Right, the shares of Common Stock owned by UCAR (rounded to the
nearest whole share) shall represent the same percentage of the aggregate number
of shares of Common Stock outstanding on a fully diluted basis as was owned by
UCAR immediately prior to such Issuance to such Recipient.

         3.2      Exercise. The Preemptive Right may be exercised by UCAR at any
time by giving written notice to the Company within 30 days after the date on
which UCAR receives

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<PAGE>   8
notice from the Company of the proposed Issuance, and the closing of the
purchase and sale pursuant to the exercise of the Preemptive Right shall occur
not sooner than 30 days after the Company receives notice of the exercise of the
Preemptive Right and prior to or concurrently with the closing of the Issuance
to the Recipient. Notwithstanding the foregoing, the Preemptive Right shall not
apply to any Issuance to all holders of shares of Common Stock on a pro rata
basis.

SECTION 4.        REGISTRATION RIGHTS

         4.1      Demand Registration.

                  (a)      Upon the written request from time to time of UCAR
that the Company effect the registration under the Securities Act of all or part
of the shares of Common Stock owned by UCAR, the Company will as expeditiously
as reasonably practicable use its reasonable efforts to effect the registration
under the Securities Act of such shares of Common Stock and cause such
registration to remain effective for a period of not less than 120 days;
provided, however, that the Company shall not be required to effect more than
two registrations pursuant to this Section 4.1 during any fiscal year of the
Company.

                  (b)      The obligations of the Company to use its reasonable
efforts to cause shares of Common Stock to be registered under the Securities
Act pursuant to this Section 4.1 are subject to the limitation that the Company
shall be entitled to postpone for a reasonable period of time (not to exceed 90
days) the filing or effectiveness of, or suspend for a reasonable period of time
(not to exceed 90 days) the rights of UCAR to make sales pursuant to, any
registration statement otherwise required to be prepared, filed and made and
kept effective by it hereunder if there has been a determination made in good
faith by the Board, in view of the advisability of deferring public disclosure
of material corporate developments or other information, that to do so would be
in the best interest of the Company at such time. The Company shall extend the
period during which such registration statement shall be maintained effective as
provided in Subsection (a) by a number of days equal to the number of days in
the period commencing on and including the date of suspension of the rights of
UCAR to make sales pursuant to such registration statement and ending on the
date sales can recommence.

                  (c)      The rights of UCAR pursuant to this Section 4.1 may
not be exercised if the shares of Common Stock to be sold by UCAR pursuant to
this Section 4.1 have a market value (based on the most recent closing share
price of Common Stock available as of such time) of less than $10,000,000.

         4.2      Piggy-Back Rights.

                  (a)      Each time the Company is planning to file a
registration statement under the Securities Act in connection with the proposed
offer and sale of Common Stock by the Company (other than in connection with an
employee stock option, purchase, ownership or other

                                       6
<PAGE>   9
plan or an acquisition, however structured, by the Company or its Subsidiaries
of a business, product line, company or assets), the Company shall give written
notice to UCAR regarding the rights of UCAR under this Section 4.2, at least 30
days prior to the anticipated filing date of such registration statement. Upon
the written request of UCAR made within 20 days after the receipt of any such
notice from the Company, which request shall specify the number of shares of
Common Stock then intended to be sold by UCAR in such offering (the "Piggy-Back
Shares"), the Company will use its reasonable efforts to effect the registration
under the Securities Act of all Piggy-Back Shares to the extent required to
permit the disposition of the Piggy-Back Shares; provided, that if, at any time
after giving written notice of its intention to register Common Stock and prior
to the effective date of the registration statement filed in connection with
such registration, the Company shall determine for any reason not to proceed
with the proposed registration, the Company may at its election give written
notice of such determination to UCAR and thereupon shall be relieved of its
obligation to register any Piggy-Back Shares in connection with such
registration only.

                  (b)      If a registration pursuant to this Section 4.2
involves an underwritten offering and the managing underwriter or underwriters
in good faith advise the Company in writing that, in their opinion, the number
of shares of Common Stock which the Company, UCAR and any other Persons intend
to include in such registration exceeds the largest number of shares of Common
Stock which can be sold in such offering without having an adverse effect on
such offering (including the price at which the shares of Common Stock can be
sold or the aggregate number of shares of Common Stock that can be sold), then
the Company will include in such registration (i) first, 100% of the shares of
Common Stock which the Company proposes to sell for its own account, if any, and
(ii) second, to the extent that the number of shares of Common Stock which the
Company proposes to sell is less than the number of shares of Common Stock which
the Company has been advised can be sold in such offering without having an
adverse effect referred to above, 100% of the Piggy-Back Shares and (iii) third,
to the extent that additional shares of Common Stock can be sold in such
offering without having an adverse effect referred to above, shares of Common
Stock sought to be included by any other Persons who are permitted to include
shares of Common Stock in such offering pursuant to any other agreements.

         4.3      Other Registration-Related Matters.

                  (a)      UCAR agrees that, if any shares of Common Stock are
offered in an underwritten Public Offering pursuant to an effective registration
statement under the Securities Act (other than a registration statement relating
to an employee stock option, purchase, ownership or other benefit plan of the
Company), it shall not, without the prior written consent of the Company, effect
any sales of Common Stock during the 14 days prior to, or the 90-day period
beginning on, the effective date of that registration statement (whether
pursuant to Section 4.1 or 4.2 or otherwise, except as part of such
registration) if and to the extent reasonably requested in writing (with
reasonable prior notice) by the managing underwriter of such underwritten Public
Offering.

                                       7
<PAGE>   10
                  (b)      The Company agrees not to effect any sales of Common
Stock during the 14 days prior to, and the 90-day period beginning on, the
effective date of any registration statement in which UCAR is selling shares of
Common Stock in an underwritten Public Offering (other than shares of Common
Stock being sold by the Company in such Public Offering), if and to the extent
reasonably requested in writing (with reasonable prior notice) by the managing
underwriter of the underwritten Public Offering.

                  (c)      The Company may require any Person that is selling
shares of Common Stock in a Public Offering pursuant to Section 4.1 or 4.2 to
furnish to the Company such information regarding such Person and the intended
distribution of such shares of Common Stock which are included in such Public
Offering as may from time to time reasonably be requested in writing in order to
comply with the Securities Act.

                  (d)      Except as otherwise provided in Section 7.4, the
Company will pay all Registration Expenses in connection with each registration
of Common Stock pursuant to this Section 4, regardless of whether such
registration becomes effective, and UCAR shall pay all underwriting discounts
and commissions and stamp, transfer and other taxes, if any, relating to the
sale of shares of Common Stock by UCAR pursuant to any registration effected
pursuant to this Section 4.

         4.4      Indemnification.

                  (a)      Indemnification by the Company. In the event of any
registration of any securities of the Company under the Securities Act pursuant
to Section 4.1 or 4.2 (which includes, for purposes of this Section 4.4, the
IPO), the Company hereby indemnifies and holds harmless, and agrees that in any
underwriting agreement entered into in connection with such registration it will
indemnify and hold harmless, to the extent permitted by law, UCAR, each
Affiliate of UCAR and their respective directors, officers, employees and
general and limited partners (and the directors, officers, , employees,
affiliates and controlling Persons thereof), each Person who participates as an
underwriter in the offering or sale of securities covered by registration
statement offered or sold by UCAR and each other Person, if any, who controls
UCAR or any such underwriter within the meaning of the Securities Act
(collectively, the "Indemnified Parties"), against any and all losses, claims,
damages or liabilities, joint or several, and expenses to which such Indemnified
Party may become subject under the Securities Act, common law or otherwise,
insofar as such losses, claims, damages or liabilities (or actions or
proceedings in respect thereof, whether or not such Indemnified Party is a party
thereto) arise out of or are based upon (i) any untrue statement or alleged
untrue statement of any material fact contained in any registration statement
relating to such registration, any preliminary, final or summary prospectus
contained therein, or any amendment or supplement thereto, or (ii) any omission
or alleged omission to state therein a material fact required to be stated
therein or necessary to make the statements therein not misleading in the light
of the circumstances under which they were made, and the Company will reimburse
such Indemnified Party for any legal or other expenses reasonably incurred by it
in connection with investigating or defending

                                       8
<PAGE>   11
any such loss, claim, damage, liability, action or proceeding; provided, that
the Company shall not be liable to any Indemnified Party in any such case to the
extent that any such loss, claim, damage, liability, action, proceeding or
expense arises out of or is based upon any untrue statement or alleged untrue
statement or omission or alleged omission made in such registration statement,
in any such preliminary, final or summary prospectus, or in any amendment or
supplement thereto, in reliance upon and in conformity with written information
with respect to such Indemnified Party furnished to the Company by such
Indemnified Party expressly for use in the preparation thereof; and provided,
further, that the Company will not be liable to any Person who participates as
an underwriter in the offering or sale of such securities or any other Person,
if any, who controls such underwriter within the meaning of the Securities Act,
under the indemnity agreement in this Section 4.4, with respect to any
preliminary prospectus, final prospectus or final prospectus as amended or
supplemented, as the case may be, to the extent that any such loss, claim,
damage or liability of such underwriter or controlling Person results from the
fact that such underwriter sold such securities to a Person to whom there was
not sent or given, at or prior to the written confirmation of such sale, a copy
of the final prospectus or of the final prospectus as then amended or
supplemented, whichever is most recent, if the Company has previously furnished
copies thereof to such underwriter. Such indemnity shall remain in full force
and effect regardless of any investigation made by or on behalf of such
Indemnified Party and shall survive the Transfer of such securities by UCAR.

                  (b)      Indemnification by UCAR and Underwriters. The Company
may require, as a condition to including any Common Stock in any registration
statement filed in accordance with this Section 4, that the Company shall have
received an undertaking reasonably satisfactory to it from the UCAR Party (as
defined below) or any such underwriter to indemnify and hold harmless (in the
same manner and to the same extent as set forth in Subsection (a), except that
the indemnification obligation of the underwriters shall be several and not
joint) the Company and all other selling stockholders, as the case may be, and
any of their respective affiliates, directors, officers, employees and
controlling Persons, with respect to any untrue statement or alleged untrue
statement in or omission or alleged omission from such registration statement,
any preliminary, final or summary prospectus contained therein, or any amendment
or supplement thereto, if such untrue statement or alleged untrue statement or
omission or alleged omission was made in reliance upon and in conformity with
written information with respect to the UCAR Party or underwriter furnished to
the Company by the UCAR Party or underwriter expressly for use in the
preparation of such registration statement, preliminary, final or summary
prospectus or amendment or supplement, or a document incorporated by reference
into any of the foregoing. Such indemnity shall remain in full force and effect
regardless of any investigation made by or on behalf of the Company or any of
the selling stockholders, or any of their respective affiliates, directors,
officers, employees or controlling Persons, and shall survive the Transfer of
such securities by the UCAR Party. The liability of any UCAR Party for any
indemnification under this Section 4.4 shall be limited to an amount equal to
the net proceeds (after deducting any underwriters' commission or discount)
received by such UCAR Party in connection with

                                       9
<PAGE>   12
the sale of securities pursuant to such registration statement.

                  (c)      Notices of Claims, Etc. Promptly after receipt by an
Indemnified Party hereunder of written notice of the commencement of any action
or proceeding with respect to which a claim for indemnification may be made
pursuant to this Section 4.4, such Indemnified Party will, if a claim in respect
thereof is to be made against an indemnifying party, give written notice to the
latter of the commencement of such action; provided, that the failure of such
Indemnified Party to give notice as provided herein shall not relieve the
indemnifying party of its obligations under such subsections (a) or (b) of this
Section 4.4, except to the extent that the indemnifying party is actually
prejudiced by such failure to give notice. In case any such action or proceeding
is brought against an Indemnified Party, unless in such Indemnified Party's
reasonable judgment a conflict of interest between the indemnified and
indemnifying parties may exist in respect of the claim to which such action or
proceeding relates and separate counsel is not employed as described below, the
indemnifying party will be entitled to participate in and to assume the defense
thereof, jointly with any other indemnifying party similarly notified, to the
extent that it may wish, with counsel reasonably satisfactory to such
Indemnified Party, and after notice from the indemnifying party to such
Indemnified Party of its election so to assume the defense thereof, the
indemnifying party will not be liable to such Indemnified Party for any legal or
other expenses subsequently incurred by the latter in connection with the
defense thereof other than reasonable costs of investigation. If, in such
Indemnified Party's reasonable judgment, having common counsel would result in a
conflict of interest between the interests of such indemnified and indemnifying
parties, then such Indemnified Party may employ separate counsel reasonably
acceptable to the indemnifying party to represent or defend such Indemnified
Party in such action, it being understood, however, that the indemnifying party
shall not be liable for the reasonable fees and expenses of more than one
separate firm of attorneys at any time for all such indemnified parties (and not
more than one separate firm of local counsel at any time for all such
indemnified parties) in such action. No indemnifying party will consent to entry
of any judgment or enter into any settlement which does not include as an
unconditional term thereof the giving by the claimant or plaintiff to the
Indemnified Parties of a release from all liability in respect of such claim or
litigation. An indemnifying party shall not be liable for any settlement of any
action, proceeding or claim effected without its prior written consent, which
shall not be unreasonably withheld.

                  (d)      Other Indemnification. Indemnification similar to
that specified in this Section 4.4 (with appropriate modifications) shall be
given by the Company and each UCAR Party with respect to any required
registration or other qualification of securities under any federal or state law
or regulation or governmental authority other than the Securities Act.

                  (e)      Contribution. If recovery is not available under the
foregoing indemnification provisions of this Section 4.4 for any reason other
than as expressly specified therein, the parties entitled to indemnification by
the terms thereof shall be entitled to contribution to liabilities and expenses
except to the extent that contribution is not permitted under

                                       10
<PAGE>   13
Section 11(f) of the Securities Act. In determining the amount of contribution
to which the respective parties are entitled, there shall be considered the
relative benefits received by each party from the offering (taking into account
the portion of the proceeds realized by each), the parties' relative fault in
connection with the statements or omissions which resulted in losses, claims,
damages or liabilities, including the parties' knowledge and access to
information concerning the matter with respect to which the claim was asserted,
the opportunity to correct and prevent any misstatement or omission and any
other equitable considerations appropriate under the circumstances.

                  (f)      Non-Exclusivity. The obligations of the parties under
this Section 4.4 shall be in addition to any liability which any party may
otherwise have to any other party. If the underwriting agreement relating to any
offering covered by this Section 4.4 provides for indemnification and
contribution of the type described in this Section 4.4, then such provisions
shall supersede and replace this Section 4.4 (other than this Subsection (f))
insofar as it specifically provides for indemnification or contribution between
the same Persons with respect to such offering. An underwriting agreement which
primarily provides for indemnification and contribution by UCAR and the Company,
on the one hand, and underwriters, on the other hand, (but not between the
Company, on the one hand, and such UCAR Party or Parties, on the other hand)
shall not supersede or replace this Section 4.4.

                  (g)      UCAR Party. "UCAR Party" shall mean UCAR or its
Subsidiary or Subsidiaries (other than the Company and its Subsidiaries),
whichever offers or sells securities under the relevant offering or
registration.

SECTION 5.        MANAGEMENT

         5.1      Composition of Board. Except as otherwise expressly agreed in
writing by UCAR, so long as UCAR owns or holds directly or indirectly 20% or
more of the then outstanding shares of Common Stock, the Company and UCAR shall
each take or cause to be taken any and all such actions so as to cause the
election (including nominating, recommending for election, soliciting proxies
for election, and voting or giving written consent or proxies as to all voting
securities of the Company owned or held for election) of: (a) a Board which
consists of six individuals of whom: (i) one is the chief executive of the
Company; (ii) three are nominees of UCAR (all of whom may be directors or
employees of UCAR and one of whom must qualify as an independent director of the
Company within the meaning of the listing rules of The Nasdaq Stock Market's
National Market); (iii) two of whom are independent directors within the meaning
of such listing rules who are not directors or employees of UCAR; and (iv)
collectively have qualifications which satisfies the committee and independent
director requirements of such listing rules in a timely manner; and (b) one of
UCAR's nominees designated by UCAR as Chairman of the Board of the Company; and
(c) an audit committee which satisfies the listing requirements of such listing
rules in a timely manner.

SECTION 6.        ACTIONS REQUIRING UCAR APPROVAL

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<PAGE>   14
         6.1      Specified Actions. So long as UCAR directly or indirectly owns
or holds 20% or more of the then outstanding shares of Common Stock, the Company
shall not, and shall not cause or permit any of its Subsidiaries to, take any of
the following actions, or authorize or approve (including those by the Board or
management) the taking of any of the following actions, or enter into any
contract, understanding or arrangement to take any of the following actions, or
publicly announce any intention or plan to take any of the following actions, in
each case without the express prior written consent of UCAR:

                  (a)      any issuance of any share of Common Stock (other than
shares issued pursuant to the underwriting agreement in connection with the IPO
and other than shares issued pursuant to Permitted Options) or any shares of
preferred stock of the Company (the "Preferred Stock");

                  (b)      any merger or consolidation of the Company with or
into any Person, other than a wholly-owned Subsidiary of the Company, or of any
Subsidiary of the Company with or into any Person other than the Company or any
other wholly-owned Subsidiary of the Company;

                  (c)      any sale, transfer or other disposition of stock or
other securities, debt or equity, of any Subsidiary of the Company or any grant
of or issuance by the Company or any of its Subsidiaries of any rights with
respect thereto (other than as collateral security for any debt permitted by
this Section 6.1)

                  (d)      any acquisition, lease (as lessee or lessor and
including a sale-leaseback) sale, transfer or other dispositions of any assets,
business or operations of the Company or any Subsidiary of the Company (other
than in the ordinary course of business) in a single transaction or a series of
related transactions having a value in excess of $5,000,000;

                  (e)      any declaration of any dividend or distribution with
respect to, or any redemption, repurchase or other acquisition of, any
securities of the Company or any Subsidiary of the Company (other than with
respect to a wholly-owned Subsidiary of the Company);

                  (f)      any liquidation, dissolution, commencement of
bankruptcy, insolvency or similar proceedings or consent or acquiescence
thereto, assignment for the benefit of creditors or similar actions with respect
to the Company or any Subsidiary of the Company;

                  (g)      any incurrence, refinancing or alteration of material
terms of any debt, or grant or alteration of material terms of any lien,
security interest or mortgage, or grant of guaranty of debt or assumption of
debt of another Person by the Company or any Subsidiary of the Company (other
than with respect to debt of up to $15,000,000 to be incurred as described in
the final prospectus relating to the IPO);

                  (h)      any capital expenditure in any year in excess of the
amount specified in the annual business plan/budget for that year approved by
the Board and by the directors of the Company nominated by UCAR (or, if no such
plan/budget is so approved for such year, in excess of the amount specified in
such plan/budget for the prior year with respect to

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<PAGE>   15
the prior year);

                  (i)      any change in accounting or tax principles or
policies except as required by generally accepted accounting principles or by
applicable law, rule or regulation;

                  (j)      any amendment to the Certificate of Incorporation or
By-laws of the Company;

                  (k)      any issuance of any securities convertible into or
exchange or exercisable for any Common Stock or Preferred Stock, including
options, warrants and rights (including employee stock options, stock
appreciation rights and restricted stock), or any other instrument that could be
deemed for any tax purposes to be outstanding equity or voting securities of the
Company (other than Permitted Options); or

                  (l)      any amendment, including any consent, waiver,
practice or action which has the effect of amending, the Company's Employee
Equity Incentive Plan or Outside Directors Equity Incentive Plan, or the forms
of the Non-Qualified Stock Option Agreements thereunder as filed with the
registration statement relating to the IPO, or adoption of any plan or
arrangement with a purpose or effect similar thereto.

SECTION 7.        OTHER AGREEMENTS

         7.1      Non-Solicitation. So long as UCAR directly or indirectly owns
or holds 20% or more of the then outstanding shares of Common Stock and, if
longer, for a period of one year from the Effective Date, neither the Company or
UCAR will or will permit its Subsidiaries to hire or solicit for hire the
employees of the other of them without the express written consent of the other
of them. For the purposes of the preceding sentence, the following does not
constitute solicitation under this Agreement: (i) use of an independent
employment agency so long as such agency is not directed to contact a specific
employee or employees of a specific company; and (ii) general advertisements not
targeted at a specific employee or employees of a specific company.

         7.2      Financial Information. The Company shall provide and shall
cause its Subsidiaries to provide all accounting, financial, business and other
information to UCAR when and as UCAR may request so as to enable UCAR to, among
other things, comply with orders and agreements, prepare financial statements,
budgets, projections and analysis, have financial statement audits and reviews
completed, comply with due diligence requests from existing or prospective
lenders, investors and strategic partners, comply with public reporting
obligations and perform customary investor relations activities, pursue and
defend claims of lawsuits and investigations, and any other proper purpose. The
obligations of the Company under this Section 7.2 shall not apply as to
information relating to periods after UCAR ceases to own or hold directly or
indirectly 20% or more of the then outstanding shares of Common Stock. For
purposes of this Section 7.2, the term "information" shall include copies of and
access to books, records, files, accounts, facilities and premises and
interviews and discussions with, certifications by and reviews of information
(within the meaning hereof by) directors, officers and employees of the Company
and its Subsidiaries).

                                       13
<PAGE>   16
7.3            Confidentiality.

                  (a)      General. UCAR hereby agrees that Confidential
Information to be furnished to it hereunder will be furnished to it in
connection with UCAR's investment in the Company. UCAR agrees that it will not
use such Confidential Information for the purpose of competing or interfering
with the Company or its Subsidiaries and that UCAR further agrees that it will
use the same practices and procedures to maintain the confidentiality of
Confidential Information as it uses to maintain the confidentiality of its own
confidential or proprietary information. Nothing contained herein shall prevent
disclosure of such Confidential Information: (i) to UCAR's Representatives (as
defined below) in the normal course of the performance of their duties or to any
financial institution providing credit to UCAR; (ii) to the extent required by
applicable law, rule or regulation (including complying with any oral or written
questions, interrogatories, requests for information or documents, subpoenas,
civil investigative demands or similar processes to which UCAR is subject);
provided, that UCAR gives the Company prompt notice of such request(s), to the
extent practicable, so that the Company may seek an appropriate protective order
or similar relief (and UCAR shall cooperate with such efforts by the Company,
and shall in any event make only the minimum disclosure required by such law,
rule or regulation)); or (iii) to any Person to whom UCAR is contemplating a
Transfer of shares of Common Stock (provided, that such Transfer would not be in
violation of the provisions of this Agreement and as long as such Third Party is
advised of the confidentiality thereof and agrees to be bound by a
confidentiality agreement consistent with the provisions hereof). Nothing
contained herein shall prevent the use or disclosure of Confidential Information
for the purposes described in Section 7.2 or the use (subject, to the extent
possible, to a protective order) of Confidential Information in connection with
the assertion or defense of any claim between the Company, or any of its
Subsidiaries, on the one hand, and UCAR, on the other hand.

                  (b)      Definition of Confidential Information. "Confidential
Information" means any information concerning the Company and Persons who are or
become its Subsidiaries or the financial condition, business, operations or
prospects of the Company and Persons who are or become its Subsidiaries
furnished to UCAR pursuant hereto (including by virtue of its present or former
right to nominate a director of the Company); provided, however, that the term
Confidential Information does not include information which (i) is or becomes
generally available to the public other than as a result of a disclosure by UCAR
or its directors, officers, employees, agents, counsel, investment advisers or
representatives (all such Persons being collectively called "Representatives")
in violation of this Agreement,(ii) is or was available to UCAR on a
non-confidential basis prior to its disclosure to such UCAR or its
Representatives by the Company or (iii) was or becomes available to such UCAR on
a non-confidential basis from a source other than the Company or its
Subsidiaries, provided, that such source is or was (at the time of receipt of
the relevant information) not, to the best of UCAR's knowledge, bound by a
confidentiality agreement with (or other confidentiality obligation to) the
Company or its Subsidiaries.

                                       14
<PAGE>   17
         7.4      Initial Public Offering Expenses. The Company and UCAR agree
that, they will bear the expenses of the IPO as follows: Each party will bear
its own internal costs relating to the IPO and each party will pay the
underwriting discounts and the commissions on the shares of Common Stock sold by
such party. External costs such as legal, accounting, printing and filing fees
will be borne pro rata by each party based on the gross proceeds received by
such party in the IPO.

         7.5      Third Party Beneficiary Provisions; Listing and Registration.
The Company shall and shall cause its Subsidiaries to comply with, observe and
perform all provisions, terms and agreements of which UCAR is a third party
beneficiary and take and cause to be taken such further actions as may be
necessary or appropriate to give full effect to the purposes and intent thereof.
Without limiting the foregoing, the Company register under the Securities Act
and all applicable foreign or state securities or blue sky laws, maintain such
registrations (or exemption therefrom) and maintain the listing or eligibility
for trading on all stock exchanges or markets on which the Common Stock is
listed or traded necessary or appropriate to issue options to purchase shares of
Common Stock in respect of options to purchase common stock of UCAR
International and to permit the exercise thereof and the purchase and public
resale of shares of Common Stock issued thereunder as provided in the Company's
Employee Equity Incentive Plan as in effect on the Effective Date.


SECTION 8.        MISCELLANEOUS

         8.1      Additional Securities Subject to Agreement.

                  (a)      Issuances by the Company. If additional shares of
Common Stock or other equity securities of the Company are issued by the Company
at any time during the term of this Agreement, either directly or upon the
conversion, exercise or exchange of securities of the Company exercisable for or
convertible or exchangeable into shares of Common Stock or other equity
securities of the Company, such Common Stock or other equity securities shall be
subject to this Agreement. All references to "Common Stock" in this Agreement
shall be deemed to include and refer to any other equity securities subject to
this Agreement in accordance with this subsection (a).

                  (b)      Acquisitions by UCAR. UCAR agrees that any other
shares of Common Stock which it shall hereafter acquire by means of a stock
split, stock dividend, distribution or otherwise (other than pursuant to a
Public Offering) shall be subject to the provisions of this Agreement to the
same extent as if held on the date hereof.

         8.2      Injunctive Relief. Each party acknowledges and agrees that a
violation of any of the terms of this Agreement will cause the other party
irreparable injury for which adequate remedy at law is not available.
Accordingly, it is agreed that each party shall be entitled to an injunction,
restraining order or other equitable relief to prevent breaches of the
provisions of this Agreement and to enforce specifically the terms and

                                       15
<PAGE>   18
provisions hereof in any court of competent jurisdiction in the United States or
any state thereof, in addition to any other remedy to which it may be entitled
at law or equity.

         8.3      Amendments. This Agreement may be amended, and compliance with
provisions, terms or agreements contained herein may be waived, only by a
written instrument signed by both of the Parties.

         8.4      Successors, Assigns and Transferees. The provisions of this
Agreement shall be binding upon and shall inure to the benefit of the parties
and their respective successors, permitted assigns and Transferees who become
parties hereto as contemplated by Section 2. Such permitted assigns and
Transferees shall have full rights to enforce and seek legal and other remedies
in respect of breaches of this Agreement as if a party. Any assignment of the
provisions hereof may be made by a party only upon the written consent of the
other party, except that UCAR may make such an assignment to any of its
Subsidiaries without such consent. Any assignment in violation of the prior
sentence shall be void and the Company shall not give effect to such assignment.

         8.5      Notices. All notices, requests and demands to or upon the
respective parties must be in writing (including by facsimile) to be effective
and, unless otherwise expressly provided herein, if in writing, shall be deemed
to have been duly given or made when delivered by hand, or two days after being
delivered to a recognized courier (whose stated terms of delivery are three days
or less to the destination of such notice) or, in the case of facsimile notice,
when received, addressed as follows or to such other address as may hereafter be
provided in writing to all of the parties:

                  When the Company is the intended recipient:

                           Graftech Inc.
                           11709 Madison Avenue
                           Lakewood, OH  44107
                           Attention:  CEO
                           with a copy to the General Counsel
                           Phone No:  (216) 529-3825
                           Facsimile No:  (216) 529-3713

                  When UCAR is the intended recipient:

                           UCAR International Inc.
                           3102 West End Avenue
                           Suite 1100
                           Nashville, TN 37203
                           Attention: President
                           with a copy to the General Counsel
                           Phone No: (615) 760-8227

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<PAGE>   19
                           Facsimile No: (615) 760-7691

         8.6      Integration. This Agreement contains the entire understanding
of the parties with respect to the subject matter hereof. There are no
agreements, representations, warranties, covenants or undertakings with respect
to the subject matter hereof other than those expressly set forth herein. This
Agreement supersedes all other prior agreements and understandings between the
parties with respect to such subject matter.

         8.7      Severability. If one or more of the provisions, paragraphs,
words, clauses, phrases or sentences contained herein, or the application
thereof in any circumstances, is held invalid, illegal or unenforceable in any
respect for any reason, the validity, legality and enforceability of any such
provision, paragraph, word, clause, phrase or sentence in every other respect
and of the remaining provisions, paragraphs, words, clauses, phrases or
sentences hereof shall not be in any way impaired, it being intended that all
rights, powers and privileges of the parties shall be enforceable to the fullest
extent permitted by law.

         8.8      Governing Law. This Agreement shall be governed by and
construed and enforced in accordance with the laws of the State of Delaware
without regard to the conflicts of law principles thereof. The parties executing
this Agreement hereby agree to submit to the non-exclusive jurisdiction of the
federal and state courts located in the State of Delaware in any action or
proceeding arising out of or relating to this Agreement.

         8.9      Counterparts. This Agreement may be executed in two or more
counterparts, and by different parties on separate counterparts, each of which
shall be deemed an original, but all of which shall constitute one and the same
instrument.

                                       17
<PAGE>   20
         IN WITNESS WHEREOF, each of the undersigned has executed this
Stockholder's Agreement or caused this Stockholder's Agreement to be executed on
its behalf as of the date first written above.


                                      UCAR International INC.


                                      By:
                                         ------------------------------------
                                         Name:
                                         Title:


                                      Graftech Inc.


                                      By:
                                         ------------------------------------
                                         Name:
                                         Title:


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