<PAGE>   1


                     RESTATED CERTIFICATE OF INCORPORATION

                                       OF

                        THE HOUSTON EXPLORATION COMPANY


               THE HOUSTON EXPLORATION COMPANY (the "Corporation"), a
corporation organized and existing under and by virtue of the General
Corporation Law of the State of Delaware ("DGCL"), hereby certifies as follows
pursuant to Sections 242 and 245 of the DGCL:

      FIRST:       The name of the Corporation is "The Houston Exploration 
                   Company."

      SECOND:      The Corporation was originally incorporated under the name
                   of "Brooklyn Union Exploration Company, Inc."

      THIRD:       The original Certificate of Incorporation of the Corporation
                   was filed in the Office of the Secretary of State of the
                   State of Delaware (the "Secretary of State") on December 16,
                   1985.

      FOURTH:      The Corporation filed amendments to its Certificate of
                   Incorporation with the Secretary of State on August 1, 1994
                   and September 19, 1995.

      FIFTH:       The sole stockholder of the Corporation is THEC Holdings
                   Corp. (the "Sole Stockholder"), a Delaware corporation.

      SIXTH:       The board of directors of the Corporation, in accordance
                   with Sections 242 and 245 of the DGCL, (i) adopted and
                   approved this Restated Certificate of Incorporation
                   (including the amendments to the Corporation's Certificate
                   of Incorporation effected hereby) and (ii) proposed that the
                   Sole Stockholder adopt and approve this Restated Certificate
                   of Incorporation (including the amendments to the
                   Corporation's Certificate of Incorporation effected hereby).

      SEVENTH:     The board of directors of the Sole Stockholder, in
                   accordance with Sections 242 and 245 of the DGCL, approved
                   and adopted on behalf of the Sole Stockholder this Restated
                   Certificate of Incorporation (including the amendments to
                   the Corporation's Certificate of Incorporation effected
                   hereby).

      EIGHTH:      The Sole Stockholder, in accordance with Section 228 of the
                   DGCL, approved and adopted this Restated Certificate of
                   Incorporation (including the amendments to the Corporation's
                   Certificate of Incorporation effected hereby).
<PAGE>   2
      NINTH:       This Restated Certificate of Incorporation shall become
                   effective upon its filing with the Secretary of State.

      TENTH:       Effective immediately upon the filing of this Restated
                   Certificate of Incorporation in the office of the Secretary
                   of State, each outstanding share of previously existing
                   Common Stock shall be and hereby is converted into and
                   reclassified as 2.47 shares of Common Stock.  Certificates
                   representing reclassified shares are hereby canceled and
                   upon presentation of the canceled certificates to the
                   Corporation, the holders thereof shall be entitled to
                   receive certificate(s) representing the new shares into
                   which such canceled shares have been converted.

      ELEVENTH:    The Certificate of Incorporation of the Corporation is
                   hereby amended and restated to read in its entirety as
                   follows:


                                   ARTICLE I

                                      Name

         The name of the Corporation is "The Houston Exploration Company."

                                   ARTICLE II

                     Registered Office and Registered Agent

         The registered office of the Corporation in the State of Delaware is 
located at Corporation Trust Center, 1209 Orange Street in the City of
Wilmington, County of New Castle.  The name of the registered agent of the
Corporation at such address is The Corporation Trust Company.

                                  ARTICLE III

                                    Purpose

         The purpose for which the Corporation is organized is to engage in any
lawful acts and activities for which corporations may be organized under the
General Corporation Law of the State of Delaware ("DGCL"), and the Corporation
shall have the power to perform all lawful acts and activities.





                                      -2-
<PAGE>   3
                                   ARTICLE IV

                                 Capitalization

          Section 4.01.  Authorized Capital.  (a) The total number of shares 
of stock that the Corporation shall have the authority to issue is 55,000,000
shares of capital stock, consisting of (i) 5,000,000 shares of preferred stock,
par value $0.01 per share (the "Preferred Stock"), and (ii) 50,000,000 shares
of common stock, par value $0.01 per share (the "Common Stock").

          (b)  Subject to the provisions of this Certificate of Incorporation 
and the Preferred Stock Designation (as defined below) creating any series of
Preferred Stock, the Corporation may issue shares of its capital stock from
time to time for such consideration (not less than the par value thereof) as
may be fixed by the Board of Directors of the Corporation (the "Board of
Directors"), which is expressly authorized to fix the same in its absolute
discretion subject to the foregoing conditions.  Shares so issued for which the
consideration shall have been paid or delivered to the Corporation shall be
deemed fully paid stock and shall not be liable to any further call or
assessment thereon, and the holders of such shares shall not be liable for any
further payments in respect of such shares.

          (c) The right to cumulate votes for the election of directors as 
provided in Section 214 of the DGCL shall not be granted and is hereby
expressly denied.

          Section 4.02.  Preferred Stock.   (a)  The Preferred Stock may be 
issued from time to time in one or more series.  Authority is hereby expressly
granted to and vested in the Board of Directors to authorize from time to time
the issuance of Preferred Stock in one or more series.   With respect to each
series of Preferred Stock authorized by it, the Board of Directors shall be
authorized to establish by resolution or resolutions, and by filing a
certificate pursuant to applicable law of the State of Delaware (the "Preferred
Stock Designation"), the following to the fullest extent now or hereafter
permitted by the DGCL:

          (1)     the designation of such series;

          (2)     the number of shares to constitute such series;

          (3)     whether such series is to have voting rights (full, special 
      or limited) or is to be without voting rights;

          (4)     if such series is to have voting rights, whether or not
      such series is to be entitled to vote as a separate class either alone or
      together with the holders of the Common Stock or one or more other series
      of Preferred Stock;





                                      -3-
<PAGE>   4
          (5)     the preferences and relative, participating, optional, 
      conversion or other special rights (if any) of such series and the
      qualifications, limitations or restrictions (if any) with respect to such
      series;

          (6)     the redemption rights and price(s), if any, of such
      series, and whether or not the shares of such series shall be subject to
      the operation of retirement or sinking funds to be applied to the
      purchase or redemption of such shares for retirement and, if such
      retirement or sinking funds or funds are to be established, the periodic
      amount thereof and the terms and provisions relative to the operation
      thereof;

          (7)     the dividend rights and preferences (if any) of such series,
      including, without limitation, (i) the rates of dividends payable
      thereon, (ii) the conditions upon which and the time when such dividends
      are payable, (iii) whether or not such dividends shall be cumulative or
      noncumulative and, if cumulative, the date or dates from which such
      dividends shall accumulate and (iv) whether or not the payment of such
      dividends shall be preferred to the payment of dividends payable on the
      Common Stock or any other series of Preferred Stock;

          (8)     the preferences (if any), and the amounts thereof, which
      the holders of such series shall be entitled to receive upon the
      voluntary or involuntary liquidation, dissolution or winding-up of, or
      upon any distribution of the assets of, the Corporation;

          (9)     whether or not the shares of such series, at the option
      of the Corporation or the holders thereof or upon the happening of any
      specified event, shall be convertible into or exchangeable for (i) shares
      of Common Stock, (ii) shares of any other series of Preferred Stock or
      (iii) any other stock or securities of the Corporation;

          (10)    if such series is to be convertible or exchangeable, the
      price or prices or ratio or ratios or rate or rates at which such
      conversion or exchange may be made and the terms and conditions (if any)
      upon which such price or prices or ratio or ratios or rate or rates may
      be adjusted; and

          (11)    such other rights, powers and preferences with respect
      to such series as may to the Board of Directors seem advisable.

Any series of Preferred Stock may vary from any other series of Preferred Stock
in any or all of the foregoing respects and in any other manner.

          (b)     The Board of Directors may, with respect to any existing 
series of Preferred Stock but subject to the Preferred Stock Designation
creating such series, (i) increase the number of shares of Preferred Stock
designated for such series by a resolution adding to such series authorized and
unissued shares of Preferred Stock not designated for any other series and (ii)
decrease the number of shares of Preferred Stock designated for such series by
a resolution subtracting from such series shares of Preferred Stock designated
for such series (but not below the number of shares of





                                      -4-
<PAGE>   5
such series then outstanding), and the shares so subtracted shall become
authorized, unissued and undesignated shares of Preferred Stock.

          (c)     No vote of the holders of the Common Stock or the Preferred 
Stock shall, unless otherwise expressly provided in a Preferred Stock
Designation creating any series of Preferred Stock, be a prerequisite to the
issuance of any shares of any series of the Preferred Stock authorized by and
complying with the conditions of this Certificate of Incorporation.  Shares of
any series of Preferred Stock that have been authorized for issuance pursuant
to this Certificate of Incorporation and that have been issued and reacquired
in any manner by the Corporation (including upon conversion or exchange
thereof) shall be restored to the status of authorized and unissued shares of
Preferred Stock and may be reissued as part of a new series of Preferred Stock
to be created by resolution or resolutions of the Board of Directors and a
Preferred Stock Designation as set forth above.

          Section 4.03.    Common Stock.  (a)  The holders of shares of
the Common Stock shall be entitled to vote upon all matters submitted to a vote
of the common stockholders of the Corporation and shall be entitled to one vote
for each share of the Common Stock held.

          (b)     Subject to the prior rights and preferences (if any)
applicable to shares of Preferred Stock of any series, the holders of shares of
the Common Stock shall be entitled to receive such dividends (payable in cash,
stock or otherwise) as may be declared thereon by the Board of Directors at any
time and from time to time out of any funds of the Corporation legally
available therefor.

          (c)     In the event of any voluntary or involuntary
liquidation, dissolution or winding-up of the Corporation, after payment or
provision for payment of the debts and other liabilities of the Corporation and
subject to the preferential or other rights (if any) of the holders of shares
of the Preferred Stock in respect thereof, the holders of shares of the Common
Stock shall be entitled to receive all the remaining assets of the Corporation
available for distribution to its stockholders, ratably in proportion to the
number of shares of the Common Stock held by them.  For purposes of this
paragraph (c), a liquidation, dissolution or winding-up of the Corporation
shall not be deemed to be occasioned by or to include (i) any consolidation or
merger of the Corporation with or into another corporation or other entity or
(ii) a sale, lease, exchange or conveyance of all or a part of the assets of
the Corporation.

          Section 4.04.  Stock Options, Warrants, etc.  Unless otherwise
expressly prohibited in the Preferred Stock Designation creating any series of
Preferred Stock, the Corporation shall have authority to create and issue
warrants, rights and options entitling the holders thereof to purchase from the
Corporation shares of the Corporation's capital stock of any class or series or
other securities of the Corporation for such consideration and to such persons,
firms or corporations as the Board of Directors, in its sole discretion, may
determine, setting aside from the authorized but unissued stock of the
Corporation the requisite number of shares for issuance upon the exercise of
such warrants, rights or options.  Such warrants,  rights and options shall be
evidenced by one or more instruments approved by the Board of Directors.  The
Board of Directors shall be empowered





                                      -5-
<PAGE>   6
to set the exercise price, duration, time for exercise and other terms of such
warrants, rights or options; provided, however, that the consideration to be
received for any shares of capital stock subject thereto shall not be less than
the par value thereof.


                                   ARTICLE V

                                   Directors

          Section 5.01.  Number and Term.  The number of directors of the
Corporation shall from time to time be fixed exclusively by the Board of
Directors in accordance with, and subject to the limitations set forth in, the
bylaws of the Corporation (the "Bylaws"); provided, however, that the Board of
Directors shall at all times consist of a minimum of three and a maximum of
fifteen members, subject, however, to increases above fifteen members as may be
required in order to permit the holders of any series of Preferred Stock to
exercise their right (if any) to elect additional directors under specified
circumstances.  No decrease in the number of directors shall have the effect of
shortening the term of any incumbent director.  Anything in this Certificate of
Incorporation or the Bylaws to the contrary notwithstanding, each director
shall hold office until his successor is elected and qualified or until his
earlier death, resignation or removal.

          Section 5.02.  Classification.  The Board of Directors shall be
divided into three classes designated as Class I, Class II and Class III,
respectively, all as nearly equal in number as possible, with each director
then in office receiving the classification to be  determined with respect to
such director by the Board of Directors.  The initial term of office of Class I
directors shall expire at the annual meeting of the Corporation's stockholders
in 1997.  The initial term of office of Class II directors shall expire at the
annual meeting of stockholders in 1998.  The initial term of office of Class
III directors shall expire at the annual meeting of stockholders in 1999.  Each
director elected at an annual meeting of stockholders to succeed a director
whose term is then expiring shall hold office until the third annual meeting of
stockholders after his election or until his successor is elected and qualified
or until his earlier death, resignation or removal.  Increases and decreases in
the number of directors shall be apportioned among the classes of directors so
that all classes will be as nearly equal in number as possible.

          Section 5.03.  Nomination and Election.   (a) Nominations of
persons for election or reelection to the Board of Directors may be made by or
at the direction of the Board of Directors.  The Bylaws may set forth
procedures for the nomination of persons for election or reelection to the
Board of Directors and only persons who are nominated in accordance with such
procedures (if any) shall be eligible for election or reelection as directors
of the Corporation; provided, however, that such procedures shall not infringe
upon (i) the right of the Board of Directors to nominate persons for election
or reelection to the Board of Directors or (ii) the rights of the holders of
any class or series of Preferred Stock, voting separately by class or series,
to elect additional directors under specified circumstances.





                                      -6-
<PAGE>   7
          (b)     Each director shall be elected in accordance with this
Certificate of Incorporation, the Bylaws and applicable law.  Election of
directors by the Corporation's stockholders need not be by written ballot
unless the Bylaws so provide.

          Section 5.04.  Removal.  No director of any class may be removed
before the expiration of his term of office except for cause and then only by
the affirmative vote of the holders of not less than a majority in voting power
of all the outstanding shares of capital stock of the Corporation entitled to
vote generally in an election of directors, voting together as a single class.
The Board of Directors may not remove any director, and no recommendation by
the Board of Directors that a director be removed may be made to the
Corporation's stockholders unless such recommendation is set forth in a
resolution adopted by the affirmative vote of not less than two-thirds of the
whole Board of Directors.

          Section 5.05.  Vacancies.  (a)  In case any vacancy shall occur on 
the Board of Directors because of death, resignation or removal, such vacancy
may be filled only by a majority (or such higher percentage as may be specified
in the Bylaws) of the directors remaining in office (though less than a
quorum), and the director so appointed shall serve for the unexpired term of
his predecessor or until his successor is elected and qualified or until his
earlier death, resignation or removal.  If there are no directors then in
office, an election of directors may be held in the manner provided by
applicable law.

          (b)     Any newly-created directorship resulting from any increase 
in the number of directors may be filled only by a majority (or such higher
percentage as may be specified in the Bylaws) of the directors then in office
(though less than a quorum), and the director so appointed shall be assigned to
such class of directors as such majority of directors shall determine;
provided, however, that newly-created directorships shall be apportioned among
the classes of directors so that all classes will be as nearly equal in number
as possible.  Each director so appointed shall hold office for the remaining
term of the class to which he is assigned or until his successor is elected and
qualified or until his earlier death, resignation or removal.

          (c)     Except as expressly provided in this Certificate of
Incorporation or as otherwise provided by applicable law, stockholders of the
Corporation shall not have the right to fill vacancies on the Board of
Directors, including newly-created directorships.

          Section 5.06.  Subject to Rights of Holders of Preferred Stock.
Notwithstanding the foregoing provisions of this Article V, if the Preferred
Stock Designation creating any series of Preferred Stock entitles the holders
of such Preferred Stock, voting separately by class or series, to elect
additional directors under specified circumstances, then all provisions of such
Preferred Stock Designation relating to the nomination, election, term of
office, removal, filling of vacancies and other features of such directorships
shall, as to such directorships, govern and control over any conflicting
provisions of this Article V, and such directors so elected need not be divided
into classes pursuant to this Article V unless expressly provided by the
provisions of such Preferred Stock Designation.





                                      -7-
<PAGE>   8
          Section 5.07.  Limitation of Personal Liability.  (a) No person who 
is or was a director of the Corporation shall be personally liable to the
Corporation or its stockholders for monetary damages for breach of fiduciary
duty as a director, except for liability (i) for any breach of the director's
duty of loyalty to the Corporation or its stockholders, (ii) for acts or
omissions not in good faith or which involve intentional misconduct or a
knowing violation of law, (iii) under Section 174 of the DGCL or (iv) for any
transaction from which the director derived an improper personal benefit.

          (b)     If the DGCL is hereafter amended to authorize corporate
action further limiting or eliminating the personal liability of directors,
then the personal liability of the directors to the Corporation or its
stockholders shall be limited or eliminated to the full extent permitted by the
DGCL, as so amended from time to time.


                                   ARTICLE VI

                              Amendment of Bylaws

          The Board of Directors is expressly authorized and empowered to
adopt, alter, amend or repeal the Bylaws.  Stockholders of the Corporation
shall have the power to alter, amend, expand or repeal the Bylaws but only by
the affirmative vote of the holders of not less than 66-2/3% in voting power of
all outstanding shares of capital stock of the Corporation entitled to vote
generally at an election of directors, voting together as a single class.


                                  ARTICLE VII

                     Actions and Meetings of  Stockholders

          Section 7.01.    No Action by Written Consent.  No action shall
be taken by the stockholders of the Corporation except at an annual or special
meeting of stockholders.  Stockholders of the Corporation may not act by
written consent in lieu of a meeting.

          Section 7.02.    Meetings.  (a) Meetings of the stockholders of
the Corporation (whether annual or special) may only be called by the Board of
Directors or by such officer or officers of the Corporation as the Board of
Directors may from time to time authorize to call meetings of the stockholders
of the Corporation.  Stockholders of the Corporation shall not be entitled to
call any meeting of stockholders or to require the Board of Directors or any
officer or officers of the Corporation to call a meeting of stockholders except
as otherwise expressly provided in the Bylaws or in the Preferred Stock
Designation creating any series of Preferred Stock.





                                      -8-
<PAGE>   9
          (b)     Stockholders of the Corporation shall not be entitled to
propose business for consideration at any meeting of stockholders except as
otherwise expressly provided in the Bylaws or in the Preferred Stock
Designation creating any series of Preferred Stock.

          (c)     Business transacted at any special meeting of stockholders 
shall be limited to the purposes stated in the notice or waivers of notice of
such meeting.  The person presiding at a meeting of stockholders may determine
whether business has been properly brought before the meeting and, if the facts
so warrant, such person may refuse to transact any business at such meeting
which has not been properly brought before such meeting.

          Section 7.03.    Appoint and Remove Officers, etc.  The stockholders 
of the Corporation shall have no right or power to appoint or remove officers
of the Corporation nor to abrogate the power of the Board of Directors to elect
and remove officers of the Corporation.  The stockholders of the Corporation
shall have no power to appoint or remove directors as members of committees of
the Board of Directors nor to abrogate the power of the Board of Directors to
establish one or more such committees or the power of any such committee to
exercise the powers and authority of the Board of Directors.


                                  ARTICLE VIII

                   Indemnification of Directors and Officers

          The Corporation shall indemnify, to the fullest extent permitted by 
applicable law and pursuant to the Bylaws, each person who is or was a director
or officer of the Corporation, and may indemnify each employee and agent of the
Corporation and all other persons whom the Corporation is authorized to
indemnify under the provisions of the DGCL.


                                   ARTICLE IX

               Election to be Governed by Section 203 of the DGCL

          The Corporation hereby elects to be governed by Section 203 of
the DGCL; provided, however, that the provisions of this Article IX shall not
apply to restrict a business combination between the Corporation and an
interested stockholder (as defined in Section 203 of the DGCL) of the
Corporation if either (i) such business combination was approved by the Board
of Directors prior to the time that such stockholder became an interested
stockholder or (ii) such stockholder became an interested stockholder as a
result of, and at or prior to the effective time of, a transaction which was
approved by the Board of Directors prior to the time that such stockholder
became an interested stockholder.





                                      -9-
<PAGE>   10
                                   ARTICLE X

                   Amendment of Certificate of Incorporation

          The Corporation reserves the right to amend, alter, change or repeal 
any provisions contained in this Certificate of Incorporation, in the manner
now or hereafter prescribed by applicable law, and all rights conferred upon
stockholders, directors or any other persons by or pursuant to this Certificate
of Incorporation are granted subject to this reservation. Notwithstanding the
foregoing or any other provision of this Certificate of Incorporation or any
provision of law that might otherwise permit a lesser or no vote, the
provisions of this Article X and of Articles V, VI and VII may not be repealed
or amended in any respect, and no provision inconsistent with any such
provision or imposing cumulative voting in the election of directors may be
added to this Certificate of Incorporation, unless such action is approved by 
the affirmative vote of the holders of not less than 66-2/3% in voting power of
all outstanding shares of capital stock of the Corporation entitled to vote
generally at an election of directors, voting together as a single class;
provided, however, that any amendment or repeal of Section 5.07 or Article VIII
of this Certificate of Incorporation shall not adversely affect any right or
protection existing thereunder in respect of any act or omission occurring
prior to such amendment or repeal and, provided further, that no Preferred
Stock Designation shall be amended after the issuance of any shares of the
Series of Preferred Stock created thereby, except in accordance with the terms
of such Preferred Stock Designation and the requirements of applicable law.


                                   ARTICLE XI

                       Voting Requirements Not Exclusive

          The voting requirements contained in this Certificate of 
Incorporation shall be in addition to the voting requirements imposed by law or
by the Preferred Stock Designation creating any series of Preferred Stock.

          IN WITNESS WHEREOF, this Restated Certificate of Incorporation has 
been executed for and on behalf and in the name of the Corporation by its
officers thereunto duly authorized on September 18, 1996.



                                /s/ JAMES G. FLOYD
                                ---------------------------------------------
                                James G. Floyd, President


                                /s/ JAMES F. WESTMORELAND
                                ---------------------------------------------
                                James F. Westmoreland, Secretary






                                      -10-
<PAGE>   11
                                                                    EXHIBIT 3.1



                            CERTIFICATE OF AMENDMENT
                                     TO THE
                     RESTATED CERTIFICATE OF INCORPORATION
                                       OF
                        THE HOUSTON EXPLORATION COMPANY


        THE HOUSTON EXPLORATION COMPANY (the "Corporation"), a corporation
organized and existing under and by virtue of the General Corporation Law of
the State of Delaware ("DGCL"), hereby certifies:

        FIRST:  That in lieu of a meeting and vote of directors, the Board of
Directors of the Corporation, by unanimous written consent filed with the
minutes of proceedings of the Board of Directors of the Corporation in
accordance with the provisions of Section 141(f) of the DGCL, adopted
resolutions approving and declaring advisable the following amendment to the
Certificate of Incorporation of the Corporation:

        NOW, THEREFORE, BE IT RESOLVED, that Article V of the Restated
Certificate of Incorporation of the Corporation be amended to read in its
entirety as follows:

                                   "ARTICLE V

                                   Directors

        Section 5.01.  Number.  The number of directors of the Corporation
shall from time to time be fixed exclusively by the Board of Directors in
accordance with, and subject to the limitations set forth in, the bylaws of the
Corporation (the "Bylaws"); provided, however, that the Board of Directors
shall at all times consist of a minimum of three and a maximum of fifteen
members, subject, however, to increases above fifteen members as may be
required in order to permit the holders of any series of Preferred Stock to
exercise their right (if any) to elect additional directors under specified
circumstances.

        Section 5.02.  Term.  All of the directors of the Corporation shall be
elected to serve for the term set forth in the Bylaws and, anything in this
Certificate of Incorporation or the Bylaws to the contrary notwithstanding,
each director shall hold office until his successor is elected and qualified or
until his earlier death, resignation or removal.  Any director elected or
appointed to fill a vacancy shall hold office for the remaining term.  No
decrease in the number of directors constituting the Corporation's Board of
Directors shall shorten the term of any incumbent director.

        Section 5.03.  Nomination and Election.   (a) Nominations of persons
for election or reelection to the Board of Directors may be made by or at the
direction of the Board of Directors.  The Bylaws may set forth procedures for
the nomination of persons for election or reelection to the Board of Directors
and only persons who are nominated in accordance with such procedures (if any)
shall be eligible for election or reelection as directors of the Corporation;
provided, however, that such procedures shall not infringe upon (i) the right
of the Board of Directors to nominate persons for election or reelection to the
Board of Directors or (ii) the rights of the holders of any class or series of
Preferred Stock, voting separately by class or series, to elect additional
directors under specified circumstances.
<PAGE>   12
        (b)     Each director shall be elected in accordance with this
Certificate of Incorporation, the Bylaws and applicable law.  Election of
directors by the Corporation's stockholders need not be by written ballot
unless the Bylaws so provide.

        Section 5.04.  Removal.  No director of any class may be removed before
the expiration of his term of office except for cause and then only by the
affirmative vote of the holders of not less than a majority in voting power of
all the outstanding shares of capital stock of the Corporation entitled to vote
generally in an election of directors, voting together as a single class. The
Board of Directors may not remove any director, and no recommendation by the
Board of Directors that a director be removed may be made to the Corporation's
stockholders unless such recommendation is set forth in a resolution adopted by
the affirmative vote of not less than two-thirds of the whole Board of
Directors.

        Section 5.05.  Vacancies.  (a)  In case any vacancy shall occur on the
Board of Directors because of death, resignation or removal, such vacancy may
be filled only by a majority (or such higher percentage as may be specified in
the Bylaws) of the directors remaining in office (though less than a quorum),
and the director so appointed shall serve for the unexpired term of his
predecessor or until his successor is elected and qualified or until his
earlier death, resignation or removal.  If there are no directors then in
office, an election of directors may be held in the manner provided by
applicable law.

        (b)     Any newly-created directorship resulting from any increase in 
the number of directors may be filled only by a majority (or such higher
percentage as may be specified in the Bylaws) of the directors then in office
(though less than a quorum).  Each director so appointed shall hold office for
the remainder of the term or until his successor is elected and qualified or
until his earlier death, resignation or removal.

        (c)     Except as expressly provided in this Certificate of
Incorporation or as otherwise provided by applicable law, stockholders of the
Corporation shall not have the right to fill vacancies on the Board of
Directors, including newly-created directorships.

        Section 5.06.  Subject to Rights of Holders of Preferred Stock.
Notwithstanding the foregoing provisions of this Article V, if the Preferred
Stock Designation creating any series of Preferred Stock entitles the holders
of such Preferred Stock, voting separately by class or series, to elect
additional directors under specified circumstances, then all provisions of such
Preferred Stock Designation relating to the nomination, election, term of
office, removal, filling of vacancies and other features of such directorships
shall, as to such directorships, govern and control over any conflicting
provisions of this Article V.

        Section 5.07.  Limitation of Personal Liability.  (a) No person who is
or was a director of the Corporation shall be personally liable to the
Corporation or its stockholders for monetary damages for breach of fiduciary
duty as a director, except for liability (i) for any breach of the director's
duty of loyalty to the Corporation or its stockholders, (ii) for acts or
omissions not in good faith or which involve intentional misconduct or a
knowing violation of law, (iii) under Section 174 of the DGCL or (iv) for any
transaction from which the director derived an improper personal benefit.

        (b)     If the DGCL is hereafter amended to authorize corporate
action further limiting or eliminating the personal liability of directors,
then the personal liability of the directors to the Corporation or its
stockholders shall be limited or eliminated to the full extent permitted by the
DGCL, as so amended from time to time."


                                     -2-
<PAGE>   13
        SECOND: That at the annual meeting of stockholders of the Corporation
on April 24, 1997, a two-thirds majority of the voting power of all outstanding
capital stock of the Corporation entitled to vote generally at an election of
directors voted in favor of such amendment, in accordance with the provisions
of Article X of the Company's Restated Certificate of Incorporation and Section
242 of the DGCL.

        THIRD:  That said amendment was duly adopted in accordance with the
provisions of Sections 141 and 242 of the DGCL.

        IN WITNESS WHEREOF, The Houston Exploration Company has caused this
Certificate of Amendment to its Restated Certificate of Incorporation to be
signed by its President and attested by its Secretary this 24th day of April,
1997.


Attest:                                     THE HOUSTON EXPLORATION COMPANY
                                            
                                            
/s/ JAMES F. WESTMORELAND                   By: /s/ JAMES G. FLOYD             
---------------------------------              --------------------------------
James F. Westmoreland, Secretary               James G. Floyd, President
                                            
                                            

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